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Applied Physics
CHAPTER 1
UNITS & DIMENSIONS (ch)
1.1 PHYSICAL QUANTITIES (mh)
The quantities by means of which we describe the laws of physics are called physical quantities.There are two types of physical quantities.
1. The Fundamental quantities
2. Derived quantities
1.1.1. The fundamental quantities (h)
Physical quantities which are independent of each other and cannot be further resolved into any other physical quantity are known as fundamental quantities. Or a physical quantity which can exist independently is called Fundamental physical quantity.There are seven fundamental quantities.
Ex: Every unit of length is fundamental unit (irrespective of the system to which it belongs);millimeter, centimeter, meter, kilometer etc.
Every unit of time is a fundamental unit. microsecond, millisecond, second, minute, hour, day etc are units of time. All these units are fundamental units.
Fundamental Units Symbol-
FUNDAMENTAL QUANTITY
UNITS SYMBOL
Length METRE m Mass KILOGRAM kg Time SECOND s Electric Current AMPERE A Thermodynamic Temperature KELVIN K Luminous Intensity CANDELA cd Amount Of Substance MOLE MOL.
2. Derived Quantities –
Physical quantities which depend upon fundamental quantities or which can be derived from fundamental quantities are known as derived quantities. A physical quantity which can not exist independently is called derived physical quantity. (Or) A physical quantity which is dependent or derived from any other physical quantity is called derived physical quantity.
Ex: Every unit of speed is a derived unit ; m/sec, cm/sec, km/hr etc.
Every unit of density is a derived unit; kg/m³, gr/cm³ etc.
Every unit of acceleration is a derived unit; m/sec², cm/sec², km/hr² etc.
1.1.2 UNITS (h)
Definition : Things in which quantity is measured are known as units.
Measurement of physical quantity= (Magnitude) × (Unit)
Example- A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then
(A) n size of u
(B) n u2
(C) n u1/2
(D) n 1/U
Answer : (D)
There are three types of units
1. Fundamental or base units
2. Derived units
3. Supplementary units
1.1.2.1 Fundamental or base units (sh)
Units of fundamental quantities are called fundamental units.
Characteristics of fundamental unit (ssh)
(i) They are well defined and are of a suitable size
(ii) They are easily reproducible at all places
(iii) They do not vary with temperature, time pressure etc. i.e. Invariable.
(iv) There are seven fundamental units.
Definitions of fundamental units (ssh)
1. Metre :The distance travelled by light in Vacuum in 1/299 ,792,45.The second is called 1m.
2. Kilogram :The mass of a cylinder made of platinum iridium alloy kept an international bureau of weights a measures is b defined as1kg.
3. Second : Cesium -133 atom emits electromagnetic radiation of several wavelengths. A particular radiation is selected which corresponds to the transistion between the two hyperfine levels of the ground state of Cs - 133. Each radiation has a time period of repetition of certain characteristics. The time duration in 9, 192, 631, 770 time periods of the selected transistion is defined as 1s.
4. Ampere :Suppose two long straight wires with negligible cross-section are placed parallel to each other in vacuum at a sepration of 1m and electric currents are established in the two in same direction. The wires attract each other. If equal currents are maintained in the two wires so that the force between them is 2 × 10–7 newton perimeter of the wire, then the current in any of the wires are called 1A. Here, newton is the SI unit of force.
5. Kelvin :The fraction 1/273.16 of the thermodynamic temperature of triple point of water is called 1K.
6. Mole :The amount of a substance that contains as many elementary entities (Molecules or atoms if the substance is monoatomic) as there are number of atoms in .012 kg of carbon - 12 is called a mole. This number (number of atoms in 0.012 kg of carbon-12) is called Avogadro constant.
7. Candela: The S.I. unit of luminous intensity is 1cd which is the luminous intensity of a blackbody of surface area 1/600,000m2 the temperature of freezing platinum and at a pressure of 101,325 N/m2, in the direction perpendicular to its surface.
Examples Based on Definition of Fundamental Units (ssh)
Example- A man seeing a lighting starts counting seconds until he hears thunder. He then claims to have found an approximate but simple rule that if the count of second is divided by an integer,the result directly gives in km, the distance of the lighting source. What is the integer if the velocity of sound is 330 m/s.
Sol. If n is the integer then according to the problem = dist in km.
( t in s)/n = (v)t
n = 1/v = 1/330 * 10‐3 =3
Example- In defining the standard of length we have to specify the temperature at which the measurement should be made. Are we justified in calling length a fundamental quantity if another physical.quantity,temperature, has to be specified in choosing a standard.
Sol. Yes, length is a fundamental quantity. One meter is the distance that contains 1650 763.73 wavelength of orange-red light of Kr - 86. Hence, the standard metre is independent of temperature. But the length of an object varies with temperature and is given by the relation .
Lt = L0 (1 + αt)
We usually specify the temperature at which measurement is made.
Example- Which of the following sets cannot enter into the list of fundamental quantities in any system of units
(A) Length ; mass ; velocity
(B) Length ; time ; velocity
(C) Mass ; time; velocity
(D) Length ; time, mass
Sol. [B] Since velocity =length/time i.e. in this set a quantity is dependent on the other two quantities Where as fundamental quantities are independent.
1.1.2.2 Derived units (sh)
Units of derived quantities are called derived units.Physical quantity units
Ιllustration: Volume = (length)3 m3
Speed = length/time m/s
1.1.2.3 Supplementary Units (sh)
The units defined for the supplementary quantities namely plane angle and solid angle are called
the supplementary units. The unit for plane angle is rad and the unit for the solid angle is steradian.
Note :
The supplemental quantities have only units but no dimensions.
1.1.3 Principle System of Units (h)
1 C.G.S. system [centimetre (cm) ; gram (g) and second (s)]
2 F.P.S system [foot ; pound ; second]
3 M.K.S. system [meter ; kilogram ; second]
4 S.I. (system of international) - In 1971 the international Bureau of weight and measures held its meeting and decided a system of units. Which is known as the international system of units.
Example- The acceleration due to gravity is 9.80 m/s2. What is its value in ft/s2?
Sol. Because 1 m = 3.28 ft, therefore
9.80 m/s2 = 9.80 × 3.28 ft/s2
= 32.14 ft/s2
Example- A cheap wrist watch loses time at the rate of 8.5 second a day. How much time will the watch be off at the end of a month ?
Sol. Time delay = 8.5 s/day
= 8.5 × 30 s/ (30 day)
= 255 s/month = 4.25 min/month.
1.2 DIMENSIONS (mh)
Dimensions of a physical quantity are the powers to which the fundamental quantities must beraised to represent the given physical quantity.
Ιllustration: Force (Quantity) = mass × acceleration
= mass ×(velocity)/time
= mass ×(length)/time2
= mass × length × (time)-2
So dimensions of force : 1 in mass
1 in length
–2 in time
and Dimensional formula : [MLT–2]
1.2.1 Examples Based On Dimensions (h)
Example- (a) Can there be a physical quantity which has no unit and dimensions
(b) Can a physical quantity have unit without having dimensions
Sol. (a) Yes, strain
(b) Yes, angle with units radians
Example- Fill in the blanks
(i) Three physical quantities which have same dimensions are …………............
(ii) Mention a scalar and a vector physical quantities having same dimensions..........................
Sol. (i) Work, energy, torque
(ii) Work, torque
Example- Choose the correct statement (s)
(A)all quantities may be represented dimensionally in terms of the base quantities
(B) all base quantity cannot be represented dimensionally in terms of the rest of the base quantities
(C) the dimension of a base quantity in other base quantities is always zero.
(D) the dimension of a derived quantity is never zero in any base quantity.
Sol. [A,B,C]
(B) all the fundamental base quantities are Independent of any other quantity
(C) same as above
Example- If velocity (V), time (T) and force (F) were chosen as basic quantities, find the dimensions of mass.
Sol. Dimensionally,
Force = mass × acceleration
Force = mass ×velocity/time
Mass = force x time/velocity
mass = FTV–1
Example- In a particular system, the unit of length, mass and time are chosen to be 10cm, 10gm and 0.1s respectively. The unit of force in this system will be equivalent to
(A)1/10N (B) 1N (C) 10N (D) 100 N
Sol. Dimensionally
F = MLT–2
In C.G.S system
1 dyne = 1g 1 cm (1s)–2
In new system
1x = (10g) (10 cm) (0.1s)–2
1dyne/1x=(1g/10g)x (1cm/10cm)x(10s/1s)-2
1 dyne =1/10,000 × 1x
104 dyne = 1x
10 x = 105 dyne = 1 N
x =1/10N
Example- The time dependence of physical quantity P is found to be of the form P = P0e–αt2 Where‘t’ is the time and α is some constant. Then the constant α will
(A) be dimensionless
(B) have dimensions of T–2
(C) have dimensions of P
(D) have dimensions of P multiplied by T–2
Sol. Since in ex, x is dimensionless
In, αt2 should be dimensionless
αt2 = M0L0T0
α = M0L0T–2
1.2.2 Examples Based On Applications Of Dimensions (h)
Example- To find the dimensions of physical constants, G, h, η etc.
Sol. Dimension of (Gravitational constant)
G:F=[MLT-2]=[G][M2]/L2
G= M-1L3T-2
Dimensions of h : Plank’s constant
E = hν
ML2T–2 = h .1/T
h = ML2T–1
Dimension of η : Coefficient of viscosity.
F = 6πηvr
[MLT–2] = η[LT–1][L]
η = [ML–1T–1]
1.2.3 Examples Based of Conversion of Units from One System to Another (h)
Example- Conversion of Newton to Dyne
(MKS) (C.G.S.)
Sol. Dimensional formula of F = MLT–2
1N =1kg x 1m/sec2=1000 g×100cm/sec2
=105 g.cm/sec2
1N= 105 dynes
Example- Conversion of G from SI system to C.G.S. Dimensional formula = M–1L3T–2
Sol. G = 6.67 × 10-11 x m3/kg.s2
G = 6.67 × 10-11 x (100cm)3/1000g.(1sec)2
G = 6.67 × 10-11 x (106/103) xcm3/g.sec2
G = 6.67 × 10-8 x (cm3 )/g.sec2
Example- The density of a substance is 8 g/cm3. Now we have a new system in which unit of length is 5cm and unit of mass 20g. Find the density in this new system.
Sol. In the new system ; Let the symbol of unit of length be La
and mass be Ma.
Since 5cm = 1 La 1cm =1La/5
20g = 1Ma 1g =1Ma/20
D = 8 g/cm3= (( 8x 1/20)Ma)/(1La/5)3
D = 50 Ma/(La)3
= 50 units in the new system.
1.2.4 Examples Based on Deriving New Relation (h)
Example- To derive the Einstein mass - energy relation
Sol. E = f ( m , c)
E = k MX CY
ML2T-2=MX(LT-1)Y
ML2T-2=MXLYT-Y
Comparing the coefficients
x = 1 ; y = + 2
Through experiments ; k = 1
E = mc2
Example- When a small sphere moves at low speed through a fluid, the viscous force F opposing the motion, is found experimentaly to depend on the radius ‘r’, the velocity v of the sphere and the viscosity η of the fluid. Find the force F (Stoke’s law).
Sol. F = f (η ; r ; v)
F = k . η . r. v
MLT-2 = (ML-1T-1)X (L)Y (LT-1)Z
MLT-2= MX L-X+Y+Z T-X-Z
comparing coefficients
x = 1 , –x + y + z = 1 ; – x – z = – 2
x = y = z = 1
F = kηvr
F = 6πηvr
As through experiments : k = 6π
1.3 Significant Digits (mh)
Normally decimal is used after first digit using powers of ten,
Ιllustration :3750 m will be written as 3.750 x103 m.The order of a physical quantity is expressed in power of 10 and is taken to be 1 if ≤(10)1/2 = 3.16 and 10 if > 3.16.
Ιllustration : speed of light = 3 x 108, order = 108
Mass of electron = 9.1 x 10-31 , order = 10-30
Significant digits : In a multiplication or division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity which has the minimum number of significant digit.
Ιllustration : 12.0/7.0 will have two significant digits only.
The insignificant digits are dropped from the result if they appear after the decimal point. They are replaced by zeroes if they appear to the left of two decimal point. The least significant digit is rounded according to the rules given below.
Rounding off : If the digit next to one rounded as more then 5, the digit to be rounded is increased by 1; if the digit next to the one rounded is less than 5, the digit to be rounded is left unchanged, if the digit next to one rounded is 5, then the digit to be rounded is increased by 1 if it odd and is left unchanged if it is even.
For addition and subtraction write the numbers one below the other with all the decimal points in one line now locate the first column from left that has doubtful digits. All digits right to this column are dropped from all the numbers and rounding is done to this column. The addition and subtraction is now performed to get the answer.Number of 'Significant figure' in the magnitude of a physical quantity can neither be increased nor decreased.
Ιllustration :: If we have 3.10 kg than it can not be written as 3.1 kg or 3.100 kg.
1.3.1 Examples Based on Significant Digits (h)
Example- Round off the following numbers to three significant digits
(a) 15462
(b) 14.745
(c)14.750
(d) 14.650 × 1012
Sol. (a) The third significant digit is 4. This digit is to be rounded. The digit next to it is 6 which is greater than 5. The third digit should, therefore , be increased by 1. The digits to be dropped should be replaced by zeros because they appear to the left of the decimal. Thus, 15462 becomes 15500 on rounding to three significant digits.
(b) The third significant digit in 14.745 is 7. The number next to it is less than 5. So 14.745 becomes 14.7 on rounding to three significant digits.
(c) 14.750 will become 14.8 because the digit to be rounded is odd and the digit next to it is 5.
(d) 14.650 × 1012 will become 14.6 × 1012 because the digit to be rounded is even and the digit next to it is 5.
Example- Evaluate (25.2×1374)/33.3 All the digits in this expression are significant.
Sol. We have (25.2×1374)/33.3= 1039.7838 .....
Out of the three numbers given in the expression 25.2 and 33.3 have 3 significant digits and 1374 has four. The answer should have three significant digits. Rounded 1039.7838 .... to three significant digits, it becomes 1040. Thus , we write.
(25.2×1374)/33.3= 1040
1.4 Principle of homogeneity (mh)
It states that if the dimensions of each term on both the sides of equation are same, then the physical quantity will be correct. In general, homogeneity is defined as the quality or state of being homogeneous. It also means having a uniform structure throughout. For instance, a uniform electric field (which has the same strength and the same direction at each point) would be compatible with homogeneity (all points experience the same physics). A material constructed with different constituents can be described as effectively homogeneous in the electromagnetic material domain, when interacting with a directed radiation field(light, microwave frequencies, etc.) In physics, homogeneous usually means describing a material or system that has the same properties at every point of the space; in other words, a uniform without irregularities. In physics, it also describes a substance or an object whose properties does not
vary with position. For example, an object of uniform density is sometimes described as homogeneous. Another related definition is simply a substance that is uniform in composition.
1.4.1 To check the dimensional correctness of a given physical relation (h)
Example- To check the correctness of v = u + at, using dimensions
Dimensional formula of final velocity v = [LT-1]
Dimensional formula of initial velocity u = [LT-1]
Dimensional formula of acceleration x time, at = [LT-2 x T]
= [LT-1]
Dimensions on both sides of each term is the same. Hence, the equation is dimensionally correct.
1.4.2 To convert a physical quantity from one system of units to another (h)
The value of a physical quantity can be obtained in some other system, when its value in one system is given using the method of dimensional analysis.
Measure of a physical quantity is given by X = nu,
u - size of unit,
n - numerical value of physical quantity for the chosen unit.
Let u1 and u2 be units for measurement of a physical quantity in two systems and let n1 and n2 be the numerical values of physical quantity for two units.
n1u1 = n2u2
Let a, b and c be the dimensions of physical quantity in mass, length and time
M1, L1, T1 and M2, L2, T2 are units in two systems of mass, length and time.
This equation is used to find the value of a physical quantity in the second or the new system, when its value in the first or the given system is known.
1.4.3 To establish a relation between different physical quantity (h)
The principle of homogeneity of dimensional equation is used to derive a relation between various physical quantities.
To derive a physical relation, the dependent factors of a given physical quantity is found. Assuming its dimensions in terms of these factors, the final dimensional equation is written in terms of mass, length and time. Equating the powers of M, L and T on both sides of the dimensional equation, three equations are formed by which, value of unknown powers can be calculated. By substituting these values in the equation, the real form of relation is achieved.
One illustration to establish a relation between different physical quantities.
Let us find an expression for the time period (d) of a simple pendulum. The time period t may depend upon (i) mass m of the bob of the pendulum (whose length is length l) , (iii) acceleration due to gravity g, at the place where the pendulum is responding. (v) angle of swing q.
Combining all the four factors, we get
K - dimensionless constant
Writing dimensionally,
Comparing the powers of like terms on both sides,
So, the equation becomes
1.5 Dimensional Equation (mh)
It is the term which tells about the power with which a fundamental quantity is contained in a physical quantity.
E.g., Dimensional formula of velocity is [LT-1], that of density is [ML-3]
The dimension of any physical quantity is the combination of the basic physical dimensions that compose it. Some fundamental physical dimensions are length, mass, time, and electric charge. All other physical quantities can be expressed in terms of these fundamental physical dimensions. For example, speed has the dimension length (or distance) per unit of time, and may be measured in meters per second, miles per hour, or other units. Similarly electrical current is electrical charge per unit time (flow rate of charge) and is measured in coulombs (a unit of electrical charge) per second, or equivalently, amperes. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure the physical variables. A straightforward practical consequence is that any meaningful equation (and any inequality and inequation) must have the same dimensions on the left and right sides. Checking this is the basic way of performing dimensional analysis.
Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their dimensions if any.
1.5.1 Dimensional equation is obtained when a physical quantity is equated with its dimensional formula (h)
In general, [X] = [Ma Lb Tc]
RHS represents dimensional formula of physical quantity X, whose dimensions in mass, length and time are a, b and c respectively.
1.5.2 Dimensional Formula (h)
When velocity is defined using the fundamental units of mass, length and time, we have
when there is no mass, M0 = 1 (algebraic theory of indices).
This is the dimensional formula for velocity and we can draw the following inference.
• Unit of velocity depends on the unit of length and time and is independent of mass.
• In the unit of velocity, the power of L and T are 1 and -1 respectively.
E.g., Formula for density is ML-3
Formula for force is MLT-2
1.5.3 Some of the Dimensional Formulas (h)
The dimensional formula of a physical quantity can be obtained by defining its relation with other physical quantities and then expressing these quantities in terms of mass [M], length [L] and time [T].
1.5.4 Applications of Dimensional Analysis
1. To find the unit of a given physical quantity in a given system of units
Ιllustration F = [MLT–2]
2. In finding the dimensions of physical constants or coefficients.
1.5.5 Limitations of Dimensional Analysis
• Dimensional analysis has no information on dimensionless constants.
• If a quantity is dependent on trigonometric or exponential functions, this method cannot be used.
• In some cases, it is difficult to guess the factors while deriving the relation connecting two or more physical quantities.
• This method cannot be used in an equation containing two or more variables with same dimensions.
• It cannot be used if the physical quantity is dependent on more than three unknown variables.
• This method cannot be used if the physical quantity contains more than one term, say sum or difference of two terms.
Example:- Convert a velocity of 48 kmh-1 into ms-1
Solution:- v = 48kmh-1
1 km = 1000 m, 1h = 60 x 60 s
Example:- Convert an acceleration of 48 km min-2 into ms-1
Solution:- a = 48kmmin-2
1 km = 1000 m, 1min = 60 x 60 s
1.6 Significant Digits (mh)
Normally decimal is used after first digit using powers of ten,
Ιllustration :3750 m will be written as 3.750 x103 m.The order of a physical quantity is expressed in power of 10 and is taken to be 1 if ≤(10)1/2 = 3.16 and 10 if > 3.16.
Ιllustration : speed of light = 3 x 108, order = 108
Mass of electron = 9.1 x 10-31 , order = 10-30
Significant digits : In a multiplication or division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity which has the minimum number of significant digit.
Ιllustration : 12.0/7.0 will have two significant digits only.
The insignificant digits are dropped from the result if they appear after the decimal point. They are replaced by zeroes if they appear to the left of two decimal point. The least significant digit is rounded according to the rules given below.
Rounding off : If the digit next to one rounded as more then 5, the digit to be rounded is increased by 1; if the digit next to the one rounded is less than 5, the digit to be rounded is left unchanged, if the digit next to one rounded is 5, then the digit to be rounded is increased by 1 if it odd and is left unchanged if it is even.
For addition and subtraction write the numbers one below the other with all the decimal points in one line now locate the first column from left that has doubtful digits. All digits right to this column are dropped from all the numbers and rounding is done to this column. The addition and subtraction is now performed to get the answer.Number of 'Significant figure' in the magnitude of a physical quantity can neither be increased nor decreased.
Ιllustration :: If we have 3.10 kg than it can not be written as 3.1 kg or 3.100 kg.
1.6.1 Examples Based on Significant Digits (h)
Example- Round off the following numbers to three significant digits
(a) 15462
(b) 14.745
(c)14.750
(d) 14.650 × 1012
Sol. (a) The third significant digit is 4. This digit is to be rounded. The digit next to it is 6 which is greater than 5. The third digit should, therefore , be increased by 1. The digits to be dropped should be replaced by zeros because they appear to the left of the decimal. Thus, 15462 becomes 15500 on rounding to three significant digits.
(b) The third significant digit in 14.745 is 7. The number next to it is less than 5. So 14.745 becomes 14.7 on rounding to three significant digits.
(c) 14.750 will become 14.8 because the digit to be rounded is odd and the digit next to it is 5.
(d) 14.650 × 1012 will become 14.6 × 1012 because the digit to be rounded is even and the digit next to it is 5.
Example- Evaluate (25.2×1374)/33.3 All the digits in this expression are significant.
Sol. We have (25.2×1374)/33.3= 1039.7838 .....
Out of the three numbers given in the expression 25.2 and 33.3 have 3 significant digits and 1374 has four. The answer should have three significant digits. Rounded 1039.7838 .... to three significant digits, it becomes 1040. Thus , we write.
(25.2×1374)/33.3= 1040
1.6.2 Fractional and Percentage Errors (h)
1. Absolute error
= |experimental value – standard value|
2. If Δx is the error in measurement of x, then
Fractional error =Δx/x
Percentage error =x/Δx× 100%
3. Propagation of error (Addition and Subtraction) :
Let error in x is ± Δx, and error in y is ± Δy, then the error in x + y or x – y is ± (Δx + Δy). The errors add.
4. Multiplication and Division :
Let errors in x, y, z are respectively ± Δx, ± Δy and ± Δz. Then error in a quantity f (defined as)f is obtained from the relation
Δf/f=| a |+ | b |Δy/yΔy+ | c |Δz/z
The fraction errors (with proper multiples of exponents) add. The error in f is ± Δf.
Example- In an experiment to determine acceleration due to gravity by simple pendulum, a student commit positive error in the measurement of length and 3% negative error in the measurement of time period. The percentage error in the value of g will be-
(A) 7% (B) 10%
(C) 4% (D) 3%
Sol. We know T = k
T2 == k'(L/g) g = k'(L/T2)
Δg/g× 100 =Δl/l× 100 +T/2ΔT× 100
= 1% + 2 x 3% = 7%
Hence correct answer is (A)
REVIEW QUESTIONS-
1. Prove that trigonometric function like sin θ are dimensionless.
2. Hooke’s law states that the force, F, in a spring extended by a length x is given by F = −kx. According to Newton’s second law F = ma, where m is the mass and a is the acceleration. Calculate the dimension of the spring constant k.
3. If the velocity of light c, gravitational constant G and planks constant h be chosen as fundamental units, find the value of a gram, a cm and a sec in term of new unit of mass, length and time respectively.
(Take c = 3 x 1010 cm/sec, G = 6.67 x 108 dyn cm2 /gram2 and
h = 6.6 x 10-27 erg sec)
4. If Force (F), velocity (V) and acceleration (A) are taken as the fundamental units instead of mass, length and time, express pressure and impulse in terms of F, V and A.
5. A student while doing an experiment finds that the velocity of an object varies with time and it can be expressed as equation:
v = Xt2 + Yt +Z .
If units of v and t are expressed in terms of SI units, determine the units of constants X, Y and Z in the given equation.
6. The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed ω. Considering the relation to be K = kIaωb where k is dimensionless constant. Find a and b. Moment of Inertia of a spehere about its diameter is (2/5)Mr2.
CHAPTER-02
FORCE AND MOTION (ch)
2.1 Concept of Force (mh)
A force is a push or pull (a vector quantity). Units of force of Newtons (N) or kgm/s2 . An external force is an applied force, such as kicking a ball. An internal force is a force from within an object, such as pushing on the dashboard of a car from inside the car. External forces cause motion, internal forces do not. A net force is the resultant of several forces acting in the same or different directions. Balanced forces are those that result in a net force of zero. Unbalanced forces are those that result in a net force greater than zero.
What happens when several forces act simultaneously on an object? In this case, the object accelerates only if the net force acting on it is not equal to zero. The net force acting on an object is defined as the vector sum of all forces acting on the object. (We sometimes refer to the net force as the total force, the resultant force, or the unbalanced force.) If the net force exerted on an object is zero, the acceleration of the object is zero and its velocity remains constant. That is, if the net force acting on the object is zero, the object either remains at rest or continues to move with constant velocity.When the velocity of an object is constant (including when the object is at rest), the object is said to be in equilibrium.
2.1.1 Unit of Force (h)
A force is measured in terms of the acceleration caused on a particular mass. Forces are measured in different units. Apart from the systems of units, the unit of forces are classified into two systems called : 1) Absolute system
2) Gravitational system The measurement of forces in absolute system are not dependent on gravitational force, whereas, measurement of forces in gravitational system are dependent on gravitational forces. Hence, the measure of forces in gravitational system differ on other planets compared to earth.
When a mass of 1 pound is subjected to an acceleration of 1 ft/sec2, the force applied is called 1poundal. The unit is abbreviated as pdl.
A pound force is defined as the gravitational force exerted on 1 pound mass. It is denoted as lbf. When we say that the weight of the person is 150 pounds, we actually mean 150 lbf.
Poundal and pound force are the units of measurement of forces in fps system. In metric system a force of 1 kilogram force is the force exerted on 1 kilogram mass. It is similar to pound force in fps system. It is denoted as kgf.
In cgs system a force is measured in terms of dynes. A force of 1 dyne produces an acceleration of 1cm/sec2 on a mass of 1 gram.
In SI units, the unit of force is defined as newton. This unit is named in the honor of Sir Isaac Newton who actually had given the definition of a force. One newton produces an acceleration of 1 m/sec2 on a mass of 1 kilogram. It can easily be derived that 1 newton is equal to 105 dynes. The unit of newton is abbreviated as N.
The gravitational units and absolute units are related by acceleration due to gravity. That is, 1 pound force ≈ 32 poundage and 1 kilogram force ≈ 9.81 newtons
Relationship between newton and dyne
One newton is the force that produces an acceleration of 1m/s2 in a body of mass 1kg.
1 newton = 1 kg x 1m/s2
Another such unit for force in the CGS system, is the dyne. One dyne is the force which produces an acceleration of 1cm/s2 in a body of mass 1gram.
1 dyne = 1 gm x 1cm/s2
The relation between one dyne and one Newton is
1 N = 105 Dyne
or 1 Dyne = 10-5 N
2.1.2 Equation for Force (h)
As per the basic definition of force, the fundamental equation of force is, F = m × a,
where F is the force, m is the mass of the object on which the force is applied and a is the acceleration generated.
At this stage we will define a quantity called momentum. It is a vector quantity defined as the product of mass of an object and its velocity. It is denoted by the letter p. The equation is, p = mv
When we multiply the equation of force by the time t on both sides,
F × t = m × a × t = mv, which we defined as momentum
Therefore, F × t = p or F = pt
When ‘t’ is infinitesimally small, the force becomes large and such force is called impulse force. Momentum is a conserved quantity. Momentum can not be destroyed, it only gets transferred. This concept helps us on the study of collision of objects.
2.1.3 Types of Forces (h)
We have generally defined force as an external agency that changes the velocity of an object with a specified mass. Let us see in what situations the velocity or the state of objects are changed and thereby classify the types of forces:
• Contact Force • Non Contact Force • Tensile Force
• Normal Force
1) Contact Force
When a body is moving along the surface of another body, the motion is resisted by a force called frictional force. The frictional force is generated along the contact surface of both the bodies. The magnitude of a frictional force depends upon the nature of the materials of the bodies that are in contact. The next example, which is also encountered by us in real life is the frictional force. 2) Non Contact Force
The force which exists between two bodies which are not in contact with each other is a Non Contact Force. A very obvious force we see in day to day life is the gravitational force. Any object away from the earth is pulled towards earth by the force called gravitational force. Even if an object is moving upwards, the gravitational force tends to slow down the vertical movement of the object. The specialty of gravitational force is that it is a Non contact Force. 3) Tensile Force
We already explained that weight of an object is nothing but a type of gravitational force. When an object is suspended with a string, wire or a chain, the force of the weight of the body acting down is countered by an upward force as per Newton’s third law. This force is transmitted to the item used for suspension. Such a force is called Tensile Force. The material of the string, wire or chain should have the capacity to absorb this force, else it will get snapped.
4) Normal Force
Even if, an object is not suspended and just simply resting on a surface, the force of the object on the surface is counter acted by a force in the opposite direction. This force acts in the direction that is normal to the surface. This type of force is called as normal force. It may be noted that a frictional force is proportional to the Normal Force.
2.2 Scalars and Vectors (mh)
Some physical quantities such as length, area, volume and mass can be completely described by a single real number. Because these quantities are describable by giving only a magnitude, they are called scalars. [The word scalar means representable by position on a line; having only magnitude.] On the other hand physical quantities such as displacement, velocity, force and acceleration require both a magnitude and a direction to completely describe them. Such quantities are called vectors.
Generally a scalar is defined as a physical quantity having only a magnitude, whereas a vector is said to be a physical quantity having a magnitude and a direction. For example, mass of an object does not change, no matter which coordinate system you choose to measure it. In contrast, some other physical quantities must maintain their direction to remain the same. Any change in direction does change the measured numbers. The force acting on an object is an example of such physical quantities where direction is important. A vector is represented by a directed line segment. The length of the line is directly proportional to the magnitude and the direction of the line represents the direction of the physical quantity.A vector is shown with a tail and a tip and the direction is defined at the tip.
The Cartesian coordinate system as well as polar coordinate system can be used to represent a vector. In three dimensions, a vector can be written as a linear combination of i, j, k, the unit orthogonal triad. For example,
u = uxi + uyj + uzk, ………..(1)
where ux, uy, and uz are the projections of u on any Cartesian x, y,and z axes.
If we apply the inversion operator on the coordinate system, the new coordinates become x¢ = –x, y¢ = –y, and z¢ = –z.Thus, a vector u, under inversion, becomes vector –u. However,under this inversion, vector u´v does not become –u´v.Vectors can be displaced by keeping their magnitude and direction intact. It does not change the vector. Vectors follow parallelogram (or triangle) law of addition. This can be defined in simple terms as follows for two vectors a and b. From the tip of vector a, construct a vector parallel to b. The vector from the tail of a (or origin)
to the tip of the new parallel vector b defines the vector sum s = a + b. They also follow the commutative law. Thus, a + b = b + a.
2.2.1 Definition of Vectors (h)
Vector is a quantity having length (a non negative real number) as well as direction. Vector represents the magnitude and direction of a physical quantity.
Vector Notation:
A vector can be represent using an ordered set of components in such a way that
V = (V1, V2,......,Vn) or V = < V1, V2, .......,Vn>
A three dimension vector can be express in the engineering notation as V = Vxi + Vy j + Vzk, where Vx, Vy and Vz are the components of the vector V and i, j and k are the unit vectors in the direction x, y and z respectively.
2.2.2 Unit Normal Vector (h)
Vector function is different from the scalar function. This is because, the vector function have the scalar value and direction. Normal vector is defined as the vector normal or perpendicular to the surface. Unit vector is a vector whose length is one. The unit normal vector is a vector perpendicular to the surface with a length value of one and it is denoted by “hat� n.
The unit normal vector at any point on a surface, we just imagine a vector with the length of one pointing the direction perpendicular to the surface. Such vector labeled as n^ is called unit vector because it's length is unity and perpendicular to the surface. For a closed surface, the direction of the unit vector has been resolved in closed surface. It is taken to point outward that is away from the volume enclosed by the surface.
2.2.3 Vector Analysis (h)
Vector analysis is concerned with the differentiation and integration of the vector fields. Vector analysis is also named as vector calculus. The basic object are scalar and vector fields. Gradient, divergence and curl are the important identities of vector calculus.
Vector analysis is related with the various operation of the vectors like dot product,cross product, addition of vector etc. Vector analysis is also related with double and triple integral to integrals over curves and surfaces. These results are mainly used in the problems of fluid flow and electromagnetic theory.
2.2.4 Vector Equation (h)
Vector Equations are defined by two equations that specify both the x and y coordinates of a graph using a parameter z. Vector equation of line segment and plane are written as r = r0 + tr1 and r(t) = (1 - t)r0 + tr1 respectively, where r0, r1 are the end points of vector.
Parametric Equation
x = t and y = 2t
Vector Form
Vector form can be used to describe Parametric equation in parametric form. [x ,y] = [t , 2t]
2.2.5 Vector Operations (h)
In mathematics, just as algebra, we can have some operations on vectors as follows:
1. Vector Addition 2. Vector Subtraction 3. Vector Dot Product 4. Vector Cross Product
2.2.5.1 Vector Addition (sh)
In the vector addition, we can add two or more than two vector to find the vector sum. For this, we can use the parallelogram law for vectors. If X and Y are the two vectors, then their sum i.e. X + Y can be found by placing X and Y as head to tail and draw a vector form for them.
If X = (X1, X2,.....Xn) and Y = (Y1, Y2,....Yn), then X + Y = (X1 + Y1, X2 + Y2,........,Xn + Yn)
Method of Vector Aiddition
If we want to add two vectors to each other and find the resultant vector ( R ) there are two way of doing it so.
1. Geometric Method
a. Parallelogram method
b. Head –to-tail method
2. Mathematical Method
a. Law of cosines
b. Law of sines
c. Resolving into components
2.2.5.1.1 Geometric Method (ssh)
Suppose we have two vectors namely a and b. Both have different in direction and magnitude and we would like to add these two vector each other and find the resultant vector of these two:
a) Parallelogram method:
The vectors are drawn from the same beginning point to form the adjacent sides of a parallelogram.The parallelogram is then completed by drawing parallel lines to the two vectors a and b. The diagonal drawn from the beginning of the vector to the opposite corner of the parallelogram is the vector c representing the sum of a and b.
b) Head-to-tail:
One of the vectors, a and b, is moved parallel to itself and they are drawn head-to-tail. Neither the direction nor the length of the vector is changed during this drawing. A third vector c is drawn from the tail of first vector to the head of the second vector. Vector c represents the sum of the resultant of vectors a and b.
2.2.5.1.2 Mathematical Method (sh)
As we mentioned before another way of finding the resultant vector is to use a mathematical method. When two vectors are not perpendicular to each other, the magnitude and direction of their sum are defined by making use of the law of cosines ,the law of sines or resolved into components.
a) Law of Cosines states that:
a2 = b2 + c2 – 2 b c Cos A
b2 = a2 + c2 – 2 a c Cos B
c2 = a2 + b2 – 2 a b Cos C
b) Law of sines states that:
a/sin A= b/sin B=c/sin C
2.2.5.2 Vector Dot Product (sh)
If we have two vectors X and Y and let θ be the angle between these two vectors, then the dot product or scalar product between them is given byX.Y=|x||Y|cosθ In general, if we have n components of these two vectors X and Y respectively, then the dot product is given as follows:
X.Y=∑nj=1XjYj
= X1Y1+X2Y2+......+XnYn
2.2.5.3 Vector Cross Product (sh)
Vector product of two vectors is indicated as follows:
A ×B =A .B sinθ Let us take any two vectors A =ai +bj +ck and B =xi +yj +zk
So, multiplication of these two vectors can be defined by matrices form. This is called determinant form.
A ×B =i (bz−cy)−j (az−cx)+k (ay−bx)
2.2.5.4 Vector Substraction (sh)
Vector subtraction is the special case of the vector addition. The vector X−Y can be define as X−Y=X+(−Y). So, we can say that the difference between X and Y is the sum of X and −Y. If a and b are two vectors, then a - b = a + (-b) is called difference of b from a.
2.2.6 Unit Vector (h)
A type of vector whose modulus is unity is called a unit vector. The unit vector in the direction of a is denoted by a^a^=1. The unit vectors parallel to a are ±a^ (read as ‘a cap’). Where |a| indicates the norm or length of the vector a. In three dimension coordinate system, i, j and k are the unit vectors corresponding to the x, y and z axis respectively. Unit Vector Notation Unit Vector notation is as follows:
a^=a/|a|
2.2.7 Vector Space (h)
Vector space is a mathematical structure which is formed by the collection of vectors. The space Rn is a vector space over the field R of the real numbers. The operations, addition and scalar multiplication of n - tuples are described as follows:
(x1 , x2 , ...., xn ) + (y1 ,y2 ,........, yn ) = (x 1 + y1 , x2 + y2 ,.........,xn + yn )
α(x1 , x2 , ...., xn ) = (αx1 , αx2 , ...., αxn ), where α R.
2.2.8 Zero Vector (h)
If all the components of any vector is zero, then that vector is called the zero vector. A vector whose initial and terminal points are coincident is called a zero or null or a void vector. The zero vector is denoted by $\vec o$. Vectors other than the null vector are called proper vectors. Magnitude of zero vector is zero. but, its direction is not defined.
2.3 Velocity and Acceleration (mh)
2.3.1 Velocity (h)
Velocity is the measure of the speed of the object in a specific direction. It is a vector quantity so both the magnitude and the direction of the object are required to define the velocity of the object. The speed and velocity of an object are interrelated terms.
Velocity v is a vector, with units of meters per second (m/s).Velocity indicates the rate of change of the object’s position (r) i.e., velocity tells you how fast the object’s position is changing. The magnitude of the velocity (|| v || ) indicates the object’s speed.The direction of the velocity (dir v) indicates the object’s direction of motion. The velocity at any point is always tangent to the object’s path at that point. Thus, the velocity tells you how the object is moving. In particular, the velocity tells you which way and how fast the object is moving.
To understand it better, consider the following example
An object is moving at, say, 50 m/sec in northwest direction. Here, 50 m/sec is the speed of the object in the northwest direction. And “50 m/sec” is the magnitude of the velocity while the "northwest" is the direction in which the object is moving.
2.3.1.1 Velocity Formula (sh)
As we know that velocity is speed of object in a particular direction, so the equation of velocity is:
V = displacement/time.
V = limΔt→0Δx/Δt = dx/dtm/sec
Here Δx denotes the change in the position or displacement of the object in Δt time. From this equation the velocity can be rewritten as the first derivative of the displacement of the object from its initial position.
2.3.1.2 Average Velocity (sh)
Average velocity defines the average rate of change of position of an object with respect to time, so, the average velocity depends only on the initial position and the final position of the object and doesn’t depend on the path taken by the object to reach the final position from its initial position. Mathematically, it can be defined as:
v = Δx/Δt m / sec = x2−x1/t2−t1 m / sec Where, x2 = final position of the object x1 = initial position of the object t2 = time at which object reach the distance d2 t1 = time at which the object starts from the distance d1 2.3.1.3 Constant Velocity (sh)
For the constant velocity, an object must have a constant speed with constant direction. Constant direction, here, means that the object should keep on moving on the straight line, since only in straight line the direction of the object never changes.
For example, a bike traveling at the constant speed of 45 mph on a circular track cannot have constant velocity while if the bike travels with same constant speed in a straight line it has constant velocity of 45 mph.
2.3.1.4 Initial Velocity (sh)
Initial velocity of an object is the velocity before the application of the external force. To understand this let’s consider following examples:
1. A football is hit by the player for the first time in the field.
2. A cricket ball is hit by the batsman.
In the example 1 the initial velocity of the football is zero, since it is lying on the floor, while in the example 2 initial velocity of the cricket ball is not zero since bowler delivers this ball with some velocity towards the batsman.
2.3.1.5 Final Velocity (sh)
The final velocity of an object is: v = u+a(t2−t1) m / sec
Where,
v = final velocity of the object
u = initial velocity of the object
a = acceleration of the object
t1 = start time
t2 = time at which the object reaches the velocity v
2.3.1.6 Linear Velocity (sh)
Velocity could be of two types – linear velocity and angular velocity. The linear velocity defines the motion of the object in the straight line while the angular velocity defines the motion of object in the circular direction. The linear velocity is denoted by v and angular velocity is denoted by ω then the relation between both the velocities is:
v = ωr rad / sec
Where, v = linear velocity of the object
ω = angular velocity of the object
r = radius of curvature along which object is moving
2.3.1.7 Uniform Velocity (sh)
An object is said to be moving with the uniform velocity if it is covering the equal distances in the equal time intervals without changing direction. Note:- The important point to remember here is that, in the case of uniform velocity the object should not change its direction and it should travel with the constant speed at all time.
2.3.1.8 Velocity and Speed (sh)
Velocity is the change of the position or displacement of an object with time while speed is the rate of change of distance with respect to time. However, velocity is a vector quantity while the speed is a scalar quantity. To define a scalar quantity only magnitude is required while to measure a vector quantity magnitude as well as direction is required.
2.3.2 Acceleration (h)
If the velocity changes from time to time, then the parameter which is responsible for change in velocity is called acceleration. In other words, it is defined as the rate of change of velocity with respect to time. Acceleration (a) is a vector, with units of meters per second squared (m/s2). Acceleration indicates the rate of change of the object’s velocity (v); i.e., the acceleration tells you how fast the object’s velocity is changing. The component of the acceleration that is parallel to the velocity (a) indicates the rate of change of the object’s speed. If ( a ) is parallel to the velocity, then the object is speeding up; if it is anti-parallel to the velocity, then the object is slowing down. The component of the acceleration that is perpendicular to the velocity (a) indicates the rate of change of the object’s direction. Thus, the acceleration does not tell you the object’s motion. Instead, the acceleration tells you how the object’s motion is changing. Like velocity, acceleration imparted on an object is also a vector quantity.
2.3.2.1 Types of Acceleration (sh)
1. Uniform Acceleration If the velocity of an object changes at a uniform rate, then the acceleration that causes the change in velocity is called uniform acceleration orconstant acceleration.
For example, the force of gravity imparts an acceleration uniformly which is called acceleration due to gravity.
For every second of motion under gravity, the velocity of an object changes constantly by 32ft/sec or 9.8 m/s approximately. Thus, acceleration due to gravity is an example of Uniform acceleration or Constant acceleration.
2. Instantaneous Acceleration In cases where the velocity of an object changes continuously, if we consider a small instant, the ratio of change in velocity at a given time for an infinitesimal change in time is defined as instantaneous acceleration of the object at that particular time. In terms of calculus, instantaneous acceleration is the derivative of the velocity function.
3. Negative Acceleration If the body slows down or we can say if the initial acceleration of a body decreases with time, then the body is said to have negative acceleration.
4. Magnitude of Acceleration The magnitude of acceleration is defined as the increase in velocity to the corresponding short interval of time. The magnitude of acceleration is given by
a = Increase in Velocity/Short Time Interval = ΔvΔt
The dimensional formula for the magnitude of acceleration is [LT -2].
As said earlier, an acceleration causes a change in velocity of an object, if the change is a reduction, then the acceleration is called Negative acceleration or deceleration. If we take the same example of motion of an object under gravity for a vertical motion, the acceleration due to gravity acts as a negative acceleration.
Generally, the motions of objects are classified as motion in a straight line in a given time interval or the motion may be along a curve. The most common motion along a curve is the circular motion or rotation. Therefore, the accelerations are broadly of two types :
1. Linear acceleration in case of Linear motion 2. Angular acceleration in case of Circular motion.
Any type of acceleration is a vector quantity. It has both magnitude and direction. In case of linear motions, the direction may be anything but in case of circular motion the directions are limited to two. The object may move in a clockwise direction or in a counterclockwise direction around the center of motion.
2.3.2.2 Acceleration Equation (sh)
The acceleration equation of an object in motion or finding a general formula for acceleration for Calculating acceleration is based on the application of Newton’s second law of motion. To effect an acceleration of an object, an external agency is required and that agency is defined as the force. The mathematical definition of the force is given by:
F = m×a
where F is the force on a mass m of the object, which imparts an acceleration a.
Therefore, an acceleration of a given mass varies directly with the force, whereas, for a given force, the acceleration varies inversely with the mass of the object.
2.3.2.3 Acceleration Graph (sh)
Let us consider an object moving under constant acceleration. As a function of time, it is expressed as,
a(t) = c1, where c1 is a constant.
Since the acceleration is the derivative of a velocity, the velocity function can be found by integration as,
v(t) = c1t + c2, where c2 is another constant.
Again by next integration, the displacement can be expressed as,
s(t) = c1t2 + c2t + c3, where c3 is another constant.
When we study these functions closely, the graphs of the functions of displacement, velocity and acceleration functions are parabola, straight line with constant slope and horizontal line respectively. (i) A body moving with constant positive acceleration and zero initial velocity.
(ii) A body moving with constant positive acceleration and non-zero initial velocity.
(iii) A body moving with constant negative acceleration.
(iv) A body moving with increasing acceleration.
(v) A body moving with decreasing Acceleration.
2.3.2.4 Units for Acceleration (sh)
As per the basic definition of acceleration, it is the ratio of change in velocity to change in time. As per dimensional analysis with the basic physical constants, a = velocity/time
= ((displacement/time)/time ) = LT-2.
Therefore, the unit of acceleration must be distance/ square of time. The most common units that are used to express an acceleration are, ft/s2 and m/s2.
2.3.2.5 Acceleration Formula (sh)
Consider any body moving, it will be having some acceleration, the equations of motion related to acceleration is given by
V = U+at
S = ut + 1/2at2
V2 - U2 = 2as
where U = initial velocity
V = final velocity
a = acceleration
t = time taken
S = distance covered by the body
2.4 Motion (mh)
When the body changes its position with respect to its surroundings, the body is said to be in Motion.
Examples: Football on the ground, motion of the moon around the earth, a person in a moving bus with respect to people outside the bus, bird flying in the sky are the examples of motion.
2.4.1 Distance & Displacement (h)
The minimum distance between two points is called displacement while the actual path covered is called distance. The displacement is a vector term and distance is scalar term. Distance and displacement both have SI unit as meter.
AB + BC = distance moved and AC = displacement
The effect of AB + BC is same as effect of AC.
On one round trip, distance is 2(AB + BC) while the displacement = AC + CA = 0. Hence the distance is never zero while the displacement is zero in one round trip. As we know that the rate of change of displacement is velocity similarly we have,
Speed = Distance moved/Time taken
S = d/t
where d is distance moved.
The SI unit for velocity and speed is meter/second (m/s).
The speed is scalar term and velocity is vector term.
The speed cannot be zero since distance cannot be zero while the velocity can be zero as displacement can be zero.
2.4.2 Types of Motion (h)
The types of motion are:
• Uniform motion • Non uniform motion
a) Uniform motion: When equal distance is covered in equal interval of time, the motion is said to be in uniform motion. The bodies moving with constant speed or velocity have uniform motion or increase at the uniform rate.
b) Non Uniform motion: When unequal distances are covered in equal interval of time, the motion is said to be in non uniform motion. The bodies executing non uniform motion have varying speed or velocity.
We can even classify motion into three types:
• Translatory motion • Rotatory motion • Vibratory motion
Translatory Motion In translatory motion the particle moves from one point in space to another. This motion may be along a straight line or along a curved path. They can be classified as:
1. Rectilinear Motion - Motion along a straight line is called rectilinear motion. 2. Curvilinear Motion - Motion along a curved path is called curvilinear motion.
Rotatory Motion In Rotatory motion, the particles of the body describe concentric circles about the axis of motion. Vibratory Motion In Vibratory motion the particles move to and fro about a fixed point.
2.4.3 Equation of Motion (h)
The variable quantities in a uniformly accelerated rectilinear motion are time, speed, distance covered and acceleration. Simple relations exist between these quantities. These relations are expressed in terms of equations called equations of motion.
There are three equations of motion.
1. V = U+at
2. S = ut + 1/2at2
3. V2 - U2 = 2as
where U = initial velocity
V = final velocity
a = acceleration
t = time taken
S = distance covered by the body
2.4.3.1 Derivation of Equation of Motion (sh)
2.4.3.1.1 First Equation of Motion (ssh)
Consider a particle moving along a straight line with uniform acceleration 'a'. At t=0, let the particle be at A and u be its initial velocity and when t=t, V be its final velocity.
Acceleration = change in velocity/Time
a= (v−u)/t. at = v-u
v = u+ at ........ First equation of motion.
2.4.3.1.2 Second Equation of Motion (ssh)
Average Velocity = Total distance traveled/Total time taken Average Velocity = s/t.....(1) Average Velocity can be written as (u+v)/2 Average Velocity = (u+v)/2........(2) From equations (1) and (2) s/t = (u+v)/2 .......(3) The first equation of motion is v = u + at. Substituting the value of v in equation (3) we get
s/t = (u+u+at)/2 s = (2u+at)t/2 = (2ut+at2)/2 = 2ut/2 + at2/2 Second equation of motion. Thus s = ut + at2/2 Second equation of motion.
2.4.3.1.3 Third Equation of Motion (ssh)
Third equation of Motion The first equation of motion is v = u + at. v - u = at ... (1) Average velocity = s/t ... (2) Average velocity = (u+v)/2 ... (3) From equation (2) and equation (3) we get, (u+v)/2 = s/t ... (4) Multiplying eq (1) and eq (4) we get, (v - u)(v + u) = at x 2s/t (v - u)(v + u) = 2as [We make use of the identity a2 - b2 = (a + b) (a - b)] v2 - u2 = 2as.......................... Third equation of motion.
2.4.3.2 Derivation of Equation of Motion(Graphically) (sh)
2.4.3.2.1 First Equation of Motion (ssh)
Consider an object moving with a uniform velocity u in a straight line. Let it be, given a uniform acceleration at time, t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and s is the distance covered by the object in time t. The figure
shows the velocity-time graph of the motion of the object.
Slope of the v - t graph gives the acceleration of the moving object.
Thus, acceleration = slope = AB = BC/AC = (v−u)/(t−0), a = (v−u)/t
v - u = at
v = u + at................................................................(1)
2.4.3.2.2 Second Equation of Motion(Graphically) (ssh)
Let u be the initial velocity of an object and 'a' the acceleration produced in the body. The distance traveled s in time t is given by the area enclosed by the velocity-time graph for the time interval 0 to t.
Distance traveled s = area of the trapezium ABDO
= area of rectangle ACDO + area of ΔABC
= (OD x OA) + 1/2 (BC x AC)
= (t x u) + 1/2 (v - u) x t
= ut + 1/2 (v - u) x t
(v = u + at by I eqn of motion; v - u = at)
S = ut + 1/2at x t
S = ut + 1/2at2.
2.4.3.2.3 Third Equation of Motion(Graphically) (ssh)
Let 'u' be the initial velocity of an object and a be the acceleration produced in the body. The distance travelled 's' in time 't' is given by the area enclosed by the v - t graph.
S = area of the trapezium OABD.
= 1/2 (b1 + b2)h
= 1/2 (OA + BD) AC
= 1/2 (u + v)t ....(1)
But we know that a = (v−u)/t
Or t = (v− u)a
Substituting the value of t in eq. (1) we get,
s = 1/2 (u+v)(v−u)/a = 1/2 (v+u)(v−u)/a
2as = (v + u)(v - u)
(v + u)(v - u) = 2as [using the identity a2 - b2 = (a + b) (a - b)]
v2 - u2 = 2as........... Third Equation of Motion
2.5 Newton’s Law of Motion (mh)
Newton has given the three laws of motion.
2.5.1 Newton's first law of motion (Law of Inertia) (h)
After Aristotle, Galileo has spent much time for observing the motion of a body. He has recognized the motion of a body occurs under some forces that describe the motion and there is any reason somehow the motion ends. So, Newton's first law was actually discovered by Galileo and perfected by Descartes (who added the crucial proviso “in a straight line”'). This law states that if the motion of a given body is not disturbed by external influences then the body moves with constant velocity. In other words:
“If an object is at rest then it will remain at rest or if it is moving along a straight line with uniform speed the it will continue to keep moving unless an external force is applied on it to change its existing state.” (Newton’s First Law of motion is also known as law of inertia).
This law defines us the force while the second law provide us the means how to measure the force. Mathematically, the displacement r of the body as a function of time t can be written-
r =r0 + v x t
where r and v are constant vectors.
Common examples of inertia in our day to day life:
1. The passengers fall forward when the bus suddenly stops. This is due to inertia of motion, the lower portion of body comes to rest but the upper portion of body continue to be in motion.
2. When we shake the branches, the fruits and leaves fall. The branches are in motion while the fruits and leaves are in rest so, they get detached.
3. The dust particles get removed when we shake the carpet. This is, because the particles are at rest while the carpet is moving, so, the particles get detached.
4. When the person jumps from the moving bus, he runs through some distance due to inertia of motion.
5. Any moving body has momentum. Mathematically, the momentum is denoted by P. It’s the product of mass and its velocity.
P = mass x velocity
P = m x v
2.5.2 Newton's second law of motion (Fundamental Law of Dynamics) (h)
Newton’s first law states that when no net force acts on an object, it stays at rest or in motion with a constant velocity. The second law tells us what happens when this force is not zero. Newton used the word “motion”' to mean what we nowadays call momentum. It is necessary to take into account since the change in the momentum in time is the net force applied on the object. The second law tells us that
“The rate of change of momentum of a body is proportional to the applied force and the change takes in the direction in which the force acts. There is a proportionality constant called mass, the proportion of the force to the acceleration, that is always constant for a given object.”
To obtain the mathematical representations, let the momentum P of a body be simply defined as the product of its mass m and its velocity v : i.e., P= mv
Since the change in the momentum in time ∆t causes a force, then F=∆P/∆t
where the F vector represents the net influence, or force, exerted on the object, whose motion is under investigation, by other objects. Then the momentum change is given by
P=mv => ∆P=m∆v
inserting this result into the Equation , we get
F=∆P/∆t=m∆v/∆t => F=m∆v/∆t => F= ma that means,
� F= ma
2.5.3 Newton's third law of motion (The Law of Reaction) (h)
Suppose, now, that there are many objects in the Universe (as is, indeed, the case). According to Newton's third law, if object b exerts a force fab on object then object a must exert an equal and opposite force fba = -fab on object b. That means b applies an action on a , then as a result of this, a applies a reaction on b . It follows that all of the forces acting in the Universe can ultimately be grouped into equal and opposite action reaction pairs. Note, incidentally, that an action and its associated reaction always act on different bodies.The third law states that
“To every action there is always an equal and opposite reaction.”
Actually, it is almost a matter of common sense. Suppose that bodies a and b constitute an isolated system. If fba = -fab then this system exerts a non-zero net force f= fab + fba on itself, without the aid of any external agency. It will, therefore, accelerate forever under its own steam.
Common examples of Newton’s third law in our day today life:
1. When a person jumps from a boat, the boat moves backwards. 2. When a bullet is fired, the gun goes backwards. 3. The huge amount of smoke downward, pushes the rocket upwards. 4. When a balloon is blown, the air rushes outward while the balloon moves backward with
the same momentum.
Worked Examples
Example- 1kg stone fall freely from rest from a bridge.What is theforce causing it to accelerate? What is its speed 4s later?How far has it fallen in this time?
Sol. a)The force causing it to fall is its weight. As it is falling with acceleration due to gravity
f=ma
f=1x 9.81
f=9.81 m/s2
b) v=u+at
v=0+9.81x4
v=39.2m/s
c) s=u+1/2at2
s= 0+1/2x9.81x42
s=78.5m
Example- A lift with its load has a mass of 2000 kg. It is supported by a steel cable. Find the tension in he cable when it:is at rest accelerates upwards uniformly at 1m/s2 move upwards at a steady speed of 1 m/s moves downwards at a steady speed of 1 m/s accelerates downwards with uniform acceleration of 1 m/s2.
Sol.
a) When at rest we can use Newton’s first law which says that the resultant force of the lift is zero. Force acting down is the lifts weight, the force acting up is the tension in the cable. These two must be equal and opposite to give a resultant force of zero. So,
W=mg=2000 x 9.81
W= 19600N
T=W
b) As the lift is accelerating upwards so T must exceed the weight mg. So the resultant acceleration force
F= T-mg
by Newton second law, F = ma, so
T-mg= ma
T-2000g=2000x1
T=19600
c) As in (a), by Newton’s first law, the resultant force on the lift must be z, so
T= mg=19600N
d)As in (c) the tension in the cable will still equal mg since the change in direction of motion does not alter the fact that there is no acceleration.
T=mg=19600N
2.6 Resolution & Composition of a force (mh)
Let the force ( F) with the direction (θ )
We can resolve this force into two components:
1‐ horizontal component( Fx ) which lies on x‐ axis
2‐ vertical component ( Fy ) which lies on y‐ axis
The horizontal component may be determined as:
Fx = F . cos θ
The vertical component may be determined as:
Fy = F .sin θ
Resolution of forces is the opposite process described in the composition of forces to determine the combined effect of force. Any force can be broken down into its horizontal and vertical components. By constructing a rectangle around the original vector, with the original vector dividing the rectangle, one can find the horizontal and vertical components of the force. This exercise illustrates the relative amount of force that is directed vertically and horizontally by the vector.
2.7 FRICTION (mh)
When a body moves or tends to move on another body, a force appears between the surfaces. This force is called the force of friction and it acts opposite to the direction of motion. Its line of action is tangential to the contacting surfaces. The magnitude of this force depends on the roughness of surfaces
There are two types of friction :
(a) Friction in un-lubricated surfaces or dry surfaces, and
(b) Friction in lubricate surfaces.
The friction that exists when one dry surface slides over another dry surface is known as dry friction and the friction.If between the two surfaces a thick layer of an oil or lubricant is introduced, a film of such lubrication forms on both the surfaces. When a surface moves on the other, in effect, it is one layer of oil moving on the other and there is no direct contact between the surfaces. The friction is greatly reduced and is known as film friction.
2.7.1 Coefficient of Friction (h)
The ratio between the maximum static frictional force and the normal reaction Rnremains constant which is known as the coefficient of static friction denoted by Greek letter µ.
Coefficient of friction=Maximum static frictional force/ Normal reaction
µ=F/Rn
The maximum angle � which the resultant reaction RN makes with the normal reaction Rn known as angle of friction. It is denoted by � .
The coefficient of friction is different for different substances and even varies for different conditions of the same two surfaces.
2.7.2 Rolling Friction (h)
The frictional resistance arises only when there is relative motion between the two connecting surfaces. When there is no relative motion between the connecting surfaces or stated plainly when one surface does not slide over the other question of occurrence of frictional resistance or frictional force does not arise.When a wheel rolls over a flat surface, there is a line contact between the two surfaces, Friction parallel to the central axis of the cylinder. On the other hand when a spherical body rolls over a flat surface, there is a point contact between the two. In both the above mentioned cases there is no relative motion of slip between the line or point of contact on the flat surface because of the rolling motion. If while rolling of a wheel or that of a spherical body on the flat surface there is no deformation of depression of either of the two under the load, it is said to be a case pure rolling.
Example- A block of Mass M is moving with a velocity v on straight surface.What is the shortest distance and shortest time in which the block can be stopped if μ is coefficient of friction
a.v2/2μg,v/μg
b. v2/μg,v/μg
c.v2/2Mg,v/μg
d none of the above
Sol. Force of friction opposes the motion
Force of friction=μN=μmg
Therefore retardation =μmg/m=μg
From v2 =u2 +2as
S=v2/2μg
From v=u+at
or t=v/μg
Example-.A body is sliding down a rough inclined plane of angle of inclination θ for which coefficient of friction varies with distance y as μ(y)=Ky where K is constant.Here y is the distance moved by the body down the plane.The net force on the body is zero at A.Find the value of constant K
a. tanθ/A
b. Acotθ
c. cottanθ/A
d. Atantanθ
Sol. The downward force=mgsinθ
The upward force=μmgcosθ
Net force f(y)=mgsinθ-μmgcosθ
=mg(sinθ-kycosθ)
at y=A, f(y)=0
0=sinθ-kAcosθ
or K=tanθ/A
Example- A horizontal force of F N is necessary to just hold a block stationary against a wall. The coefficient of friction between the block and the wall is μ. The weight of the block is
a.μF
b. F(1+μ)
c. F/μ
d none of these
Sol. Let W be the weight
Reaction force=F
Weight downward=W
weight Upward=frictional force=μr=μF
For no movement
weight Upward=Weight downward
W=μF
2.8 Banking of Roads (mh)
When vehicle go through turnings, they travel along a nearly circular arc. There must be some force to produce the required centripetal acceleration. Centripetal force is provided to the vehicle by following three ways.
• By friction only
• By banking of roads only
• By friction and banking of roads both
1) By friction:
Let a car of mass m is moving at a speed v is at horizontal circular arc of radius r.
And fl = limiting value of f = µN = µmg (N=mg)
For a safe turn without sliding,
So if m and r is fixed the speed of the vehicle should not exceed (µrg)1/2 and if v and r is fixed then coefficient of friction should be greater than v2/g.
2) By banking of roads only:
To avoid friction outer part of road is some what lifted compared to the inner part.
From figure NSinø =mv2/r and NCos ø= mg
From this we get, Tanø= v2/gr or v=(rgtanø)
At two speeds car does not slide down even if track is smooth.
3) By friction and banking of road both: If a vehicle is moving on a circular road which is rough and banked also then magnitude of N and direction plus magnitude of friction mainly depends on the speed of the vehicle V-
• f is outward if v = 0
• f is inward if v>
• f is outward if v <
• f is zero if v =
Example- A turn of radius 20m is banked for the vehicle of mass 200kg going at a speed of 10m/s. Find the direction and magnitude of frictional force acting on a vehicle if it moves with a speed a) 5 m/s b) 15 m/s assume the friction is sufficient to prevent slipping ( g = 10m/s2).
Solution:
a) The turn is banked for speed v = 10 m/s therefore
As the speed is decreased force of friction f acts upwards
Substituting
b) Here force of friction f will act downwards
Substituting v=15 m/s,
Example- The mass system is kept on sphere. Ball 1 is slightly disturbed. What is the velocity of these balls when it is making angle ‘q’ with horizontal (friction is absent everywhere).
Sol.
Since they are connected by inextensible string, therefore at any stage the velocity of both particle will be same. Let it be ‘V’. From work energy theorem:
Example-A smooth, light rod AB can rotate about a vertical axis passing through its end A. The rod is fitted with small sleeve of mass m attached to the end A by a weightless spring of length l0, stiffness k. What work must be performed to slowly get this system going and the angular velocity w?
Sol. The mass m rotates in a circle of radius l, which is extended length of the spring. Centripetal force on m = k (l- l0) = mw2l
W = change in KE of m+ energy stored in the spring
2.9 Circular Motion (mh)
Uniform Circular motion: Angular velocity w is constant throughout the motion. The magnitude of velocity also remains constant. Non-Uniform Circular motion: Angular velocity ‘w’ changes with time
2.9.1 Centripetal and Tangential Forces (h)
The forces acting on a body could be resolved into two components, one in radial direction and one in tangential direction.The force in radial direction is called CENTRIPETAL FORCE and the other is called TANGENTIAL FORCE.
In equation
First part is tangential acceleration and second is centripetal acceleration.
Example- A 0.1 Kg block is undergoing circular motion. What is the range of ‘w’ for which the particle can perform the circular motion?
Solution:
For vertical Equilibrium of body
For body to rotate in circle, net force towards center of circle must be equal to mw2r
From (1) and (2)
Now R = h tan450=h
For maximum value ‘’ the body will have tendency to move upwards. Direction of friction will be in downward direction.Similar analysis yields.
2.10 Parabolic Motion of Projectiles (mh)
A projectile is an object upon which the only force is gravity. Gravity, being a downward force, causes a projectile to accelerate in the downward direction. The force of gravity could never alter the horizontal velocity of an object since perpendicular components of motion are independent of each other. A vertical force does not effect a horizontal motion. The result of a vertical force
acting upon a horizontally moving object is to cause the object to deviate from its otherwise linear path. According to Newton's law of inertia, an object in motion in a horizontal direction would continue in its horizontal motion with the same horizontal speed and direction unless acted upon by an unbalanced horizontal force. The animation above shows a green sphere moving to the right at constant speed. The horizontal distance traveled in each second is a constant value. The red sphere undergoes a vertically accelerated motion which is typical of an object upon which only the force of gravity acts. If these two motions are combined - vertical free fall motion and constant horizontal motion - then the trajectory will be that of a parabola. An object which begins with an initial horizontal velocity and is acted upon only by the force of gravity will follow the path of the blue sphere. It will travel the same horizontal distance in each consecutive second but will fall vertically a greater distance in each consecutive second.
The factors that affect the trajectory are:
a) Angle of projection
b) Projection velocity
c) Relative height of projection
In order to analyse projectile motion, it is divided into two components, horizontal motion and vertical motion. Perpendicular components of motion are independent of each other i.e. the horizontal and vertical motions of a projectile are independent. Horizontal motion of an object has no external forces acting upon it (with the exception of air resistance but this is generally not accounted for). Due to this absence of horizontal forces, a projectile remains in motion with a constant horizontal velocity, covering equal distances over equal periods in time. Thus no horizontal acceleration is occurring. The degree of vertical velocity however, is reduced by the effect of gravity. Force of gravity acts on the initial vertical velocity of the javelin, reducing the velocity until it equals zero. A vertical velocity of zero represents the apex of the trajectory, meaning that the projectile has reached its max height. During the downward flight of the projectile, vertical velocity increases due to the effect of gravity.
2.10.1 Kinematic quantities of projectile motion (h)
In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.
Acceleration-
Since there is no acceleration in the horizontal direction velocity in horizontal direction is constant which is equal to ucosα. The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, equal to g. The components of the acceleration:
ax = 0 ,
ay = -g .
Velocity
The horizontal component of the velocity remains unchanged throughout the motion. The vertical component of the velocity increases linearly, because the acceleration is constant. At any time t, the components of the velocity:
vx =v0 cos(ø) ,
vy =v0 sin(ø) - gt .
The magnitude of the velocity (under the Pythagorean theorem):
v= {vx 2 + vy2}1/2 .
Displacement
Displacement and coordinates of parabolic throwing
At any time t, the projectile's horizontal and vertical displacement:
x = v0 t cos(ø) ,
y = v0 t sin(ø)-1/2gt2
The magnitude of the displacement:
∆r= {x2 + y2 }1/2
Time of flight
The time of flight (t) is the time it takes for the projectile to finish its trajectory.
t = d/vcosø =(vsinø +(v sinø)2 + 2gy0 )/g
if θ is 45° and y0 is 0
Angle of reach
The "angle of reach" (not quite a scientific term) is the angle (φ) at which a projectile must be launched in order to go a distance d, given the initial velocity v.
sin(2ø) = gd/v2
ø = ½ arc sin(gd/v2)
Range of a projectile
The horizontal range d of the projectile is the horizontal distance the projectile has travelled when it returns to its initial height (y = 0).
0 =v0 td sin(ø) – 1/2gtd2.
Time to reach ground:
td = 2v0 sin(ø)/g
From the horizontal displacement the maximum distance of projectile:
Note that d has its maximum value when ,which necessarily corresponds to or
Example- A rock is thrown with an initial vertical velocity component of 30 m/s and an initial horizontal velocity component of 40 m/s.
a. What will these velocity components be one second after the rock reaches the top of its path?
b. Assuming the launch and landing heights are the same, how long will the rock be in the air?
c. Assuming the launch and landing heights are the same, how far will the rock land from where it was thrown?
So: A rock is thrown with an initial vertical velocity component of 30 m/s and an initial horizontal velocity component of 40 m/s.
a.The horizontal component of the velocity remains constant, 40 m/s. The vertical component of the velocity decreases by 10 m/s every second. So at the peak, the vertical component of the velocity is zero and one second later the vertical component of the velocity is -10 m/s.
b. Since the vertical component of the velocity decreases by 10 m/s every second, it will take 3 seconds for the vertical component of the velocity to slow from 30 m/s to 0 m/s, and another 3 seconds for the rock to accelerate from 0 m/s to -30 m/s. The total time in the air is 6 seconds.
c. Since the rock is in the air for six seconds and the rock moves horizontally 40 meters each second, the range for the rock is 6 s x 40 m/s = 240 m.
Example- If a person can jump a horizontal distance of 3 m on Earth, how far could the person jump on the moon where the acceleration due to gravity is one-sixth of that on earth (1.7 m/s/s)?
Solution: The horizontal distance jumped is directly proportional to the time in the air (or above the ground since there is no air on the moon):
Dx = vxDt
The time above the ground is inversely propotional to the vertical acceleration:
Dt = Dvy / a
On the moon the gravitational acceleration is six times less than on Earth so the time above the ground will be six times more.Increasing the time by a factor of six increases the horizontal displacement by a factor of six. So the person could jump six times farther, 18 m.
Example- A brick is thrown upward from the top of a building at an angle of 25 degrees above the horizontal and with an initial speed of 15 m/s. If the brick is in the air for 3 seconds, how high is the building? (Draw a picture.)
Solution: Given:
Vi= 15m/s
Dt = 3s
sin(250) = Vy/15m/s
Vy = 6.3 m/s
Now use the y-component information to find the height:
Dy = (.5)(a)(Dt2)+(Vy)(Dt)
Dy = (.5)(-10m/s2)(3s)2+(6.3m/s)(3s)
Dy = -45m +15m
Dy = -26m
Therefore the height of the building is 26m.
2.11 Review Questions (mh)
1. A train covers 60 miles between 2 p.m. and 4 p.m. How fast was it going at 3 p.m.?
2. A box sits on a horizontal wooden board. The coefficient of static friction between the box and the board is 0.5. You grab one end of the board and lift it up, keeping the other end of the board on the ground. What is the angle between the board and the horizontal direction when the box begins to slide down the board?
3. A car is initially traveling due north at 23 m/s. (a) Find the velocity of the car after 4 s if its acceleration is due north. (b) Find the velocity of the car after 4 s if its acceleration is instead due south.
4. The car drives straight off the edge of a cliff that is 57 m high. The investigator at the scene of the accident notes that the point of impact is 130 m from the base of the cliff. How fast was the car traveling when it went over the cliff?
5. A gun moving at a speed of 30m/sec fires at an angle 30o with a velocity 150m/s relative to the gun. Find the distance between the gun and the projectile when projectile hits the ground. (g = 10 m/sec)
6. A stone is thrown at a speed of 19.6 m/sec at an angle 30o above the horizontal from a tower of height 490 meter. Find the time during which the stone will be in air. Also find the distance from the foot of the tower to the point where stone hits the ground?
7. Particle is projected with a velocity 39.2 m/sec at an angle of 30o to an inclined plane (inclined at an angle of 45o to the horizontal). Find the range on the incline (a) when it is projected upward (b) when it is projected downward.
CHAPTER-03
WORK , ENERGY,POWER (ch)
3.1 Work (mh)
Work is said to be done when a force acts on an object and the point of application of the force moves in the direction of force.
Hence, the essential condition that has to be satisfied for work to be done are:
• Some force must act on the object
• The point of application of force must move in the direction of force.
The product of the force and the distance moved measures work done.
W = F x S, Where W is the work done, F is the force applied and S is the distance covered by the moving object. Work done is a scalar quantity.
Whenever force acting on a body is able to actually move it through some distance in the direction of force, the work is said to be done by the force. The force performing work may be constant or a variable.
3.2 Work done on bodies moving on horizontal and inclined planes (mh)
3.2.1 Work done by Constant Forces (h)
An object undergoes displacements along a straight line while acted on by a force F, the angle between is q. Then work done is-
W=
a.) Work done as such has no relevance until the commenced force is maintained.
b.) Work is scalar quantity (as indicated by dot product).
c.) Unit of work is Joules 1J = 1Kgm2 /s2
d.) Dimension of work is [ML2T-2]
Other Units of Work:
3.2.2 Worked Examples (h)
Example- A block of mass M is pulled along a horizontal surface by applying a force at an angle ø with horizontal. Co-efficient of friction between block and surface is µ . If the block travels with uniform velocity, find the work done by this applied force during a displacement d of the block.
Solution: The force acting on the block is shown in figure.
As the block moves with uniform velocity the force add up to zero.
FCosq = N --------- (1)
FSinq+N= Mg ------- (2)
Solving (1) and (2)
FCosq = (Mg-FSinq)
F=
Work done by this force during a displacement d is
W = F.dCosq =
3.2.3 Workdone by a variable Force (h)
Let us consider a situation where the force is acting along the X-axis and the magnitude of the force is varying with position 'x'. Thus, as the ball moves, the magnitude of the work done by the force, on the ball, changes. The adjacent graph shows the plot of a one dimensional variable force.
Let us calculate the work done on the ball by this force. We divide the area under the graph into number of narrow strips of width 'Dx'. It is small enough to assume that F (x) is uniform in that range.
Let Fja be the average value of F (x) within the jth interval. Therefore, Dwj is the work done in the jth interval time; Dwj = FjaDx
Total work, W = Dwj = FjaDx
Geometrically, the work is equal to the area between the F(x) curve and the X-axis between the limits xi and xj.
3.2.4 Work done when the force is not along the direction of motion (h) In many cases when we pull or push an object, we find that force and displacement are not in the same direction.
Let a constant force F acting on a body produce a displacement S as shown in the figure. Let q be the angle between the directions of the force and displacement.
Displacement in the direction of the force = Component of s along AX = AC
Work done = Force x displacement in the direction of force
W = F s cos q
If the displacement s is in the direction of the force F, q = 0, cos q =1
Then, W = Fs x 1
W = Fs
If q = 90o, cos 90 = 0
Therefore, W = Fs x 0 = 0 i.e, no work is done by the force on the body.
3.2.5 Derivation of work done for a particle moving along a horizontal line (h)
In the case the resultant force F is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line.[11] The relation between the net force and the acceleration is given by the equation F = ma (Newton's second law), and the particle displacement d can be expressed by the equation
d = {v22 - v1
2 }/{2a}
The work of the net force is calculated as the product of its magnitude and the particle displacement. Substituting the above equations, one obtains:
In the general case of rectilinear motion, when the net force F is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle:
3.2.6 Work Done on Body in Inclined Planes (h)
A force F acting at the angle q moves a body from point A to point B. The distance moved in the direction of the force is given by
If the body moves in the same direction as the force the angle is 0. so Work done = Fs
When the angle is 90 then the work done is zero.
The SI units for work are Joules J (with force, F, in Newton's N and distance, s, in metres m).
Example-How much work is done when a force of 5 kN moves its point of application 600mm in the direction of the force.
Sol.
Example- Find the work done in raising 100 kg of water through a vertical distance of 3m.
Sol.The force is the weight of the water, so
3.3 Types of Work (mh)
3.3.1 Positive Work (h)
When q is acute, cos q is positive. Hence, the work done is positive.
• When a body falls freely under the force of gravity, q = 0, cos q = 1.
Hence, the work done by gravity is positive.
• When a gas in a cylinder, fitted with a movable piston, is allowed to expand, work done by the gas is positive, since the force due to the gas pressure and displacement of piston are in the same direction.
• When a spring is stretched, work done by the stretching force is positive.
3.3.2 Negative Work (h)
and when q is obtuse, cos q is negative. Hence, the work done is negative.
• When a ball is thrown up, its upward motion is opposed by gravity. The displacement is in the upward direction and the gravitational force acts in the downward direction. Here, q = 180o and cos q = -1. Therefore, the work done by gravity on a body moving upward, is negative.
• Let us push a box on a table (a rough horizontal surface). The motion of the box is opposed by the force of friction. The force of friction and the displacement of the object always make an angle 180o and hence, the work done by the frictional force is always negative.
• When a positive charge is moved closer to another positive charge, the work done by the electrostatic force of repulsion between the charges is negative.
3.3.3 Zero Work (h)
When the applied force and the displacement make an angle 90o to each other, the work done is zero. This is because cos 90 = 0.
• A person carrying a briefcase, moves on a horizontal road. Since, the gravitational force acts vertically downwards and the displacement of the person is in the horizontal direction, the work done by the man against gravity is zero.
• Take the case of an oscillating simple pendulum. The displacement of the bob at any point on the arc is along the tangent drawn at that point. The tension of the string is along the radius of the arc. Hence, the tension of the string is always perpendicular to the displacement and the work done by the tension is always zero.
3.4 Power (mh)
The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber.
Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.
The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.
Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that
particular machine. A car engine is an example of a machine that is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160-horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. The power equation suggests that a more powerful engine can do the same amount of work in less time.
A person is also a machine that has a power rating. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student's personal power. Despite the diagonal motion along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the student is equal to the force that does the work on the student and the height of the staircase is the upward displacement. Suppose that Ben Pumpiniron elevates his 80-kg body up the 2.0-meter stairwell in 1.8 seconds. If this were the case, then we could calculate Ben's power rating. It can be assumed that Ben must apply an 800-Newton downward force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben's body with just enough force to lift his body up the stairs. It can also be assumed that the angle between the force of the stairs on Ben and Ben's displacement is 0 degrees. With these two approximations, Ben's power rating could be determined as shown below.
The expression for power is work/time. And since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below.
This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity). A powerful car engine is strong and fast. A powerful piece of farm equipment is strong and fast. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A machine that is strong enough to apply a big force to cause a displacement in a small mount of time (i.e., a big velocity) is a powerful machine.
If work is being done by a machine moving at speed v against a constant force, or resistance, F, then since work doe is force times distance, work done per second is Fv, which is the same as power.
Example- A constant force of 2kN pulls a crate along a level floor a distance of 10 m in 50s.What is the power used?
Sol.
Example- A hoist operated by an electric motor has a mass of 500 kg. It raises a load of 300 kg vertically at a steady speed of 0.2 m/s. Frictional resistance can be taken to be constant at 1200 N.What is the power required?
Sol.
Example- A man of mass 60 kg runs up a flight of 60 steps in 40 seconds. If each step is 20 cm high, calculate his power.
Solution:
Since the man runs up a flight of steps, the work done will be equal to the potential energy.
Height = height of one step x number of steps
Acceleration due to gravity (g)=9.8m/s2
Time = 40 sec
Example- Calculate the power of an engine required to lift 105kg of coal per hour from a mine 100m deep. Given g=10m/s2.
Solution:
The work done in lifting coal is nothing but gravitational potential energy.
Mass of coal = 105 kg
Time = 1 hour
Power of the engine = 27777.78 W
Example- A man buys an electric motor of 2 h.p. What is its power in watts?
Solution:
Power of the electric motor = 2 h.p
Power of the electric motor = 2 x 746
3.5 Potential Energy (mh)
When you call a person a potential person, what exactly does it mean? It means that person has acquired some virtues by his deeds, inhabitation or by his position and capable of doing something great. Similarly, the Potential Energy definition of an object is better explained as the energy acquired by it by virtue of its position and an energy which is readily available for use. A potential energy in an object created by the work done by a force on it. Thus potential energy can be defined as:
“An energy acquired by an object by virtue of its position is called its potential energy.”
There are different forms of potential energy two examples are: i) a pile driver raised ready to fall on to its target possesses gravitational potential energy while (ii) a coiled spring which is compressed possesses an internal potential energy.Only gravitational potential energy will be considered here. It may be described as energy due to position relative to a standard position (normally chosen to be he earth's surface.)
The potential energy of a body may be defined as the amount of work it would do if it were to move from the its current position to the standard position.
3.5.1 Potential Energy Formula (h)
Consider an object of mass 'm', raised through a height 'h' above the earth's surface. The work done against gravity gets stored in the object as its Potential Energy (Gravitational Potential Energy).
potential energy = Work done in raising the object through a height 'h'.
Consider an object of mass 'm', raised through a height 'h', then its Potential energy is given by: Potential energy = F × h .......................(1) where, F= force, h = height attained due to its displacement. But F = mg (Newton's second law of motion) Substituting for F in equation (1) we get, Potential energy, P.E = mgh The above relation is called Potential Energy Equation. From this, it is clear that the Potential Energy of an object depends on the height from the ground.
Example- What is the potential energy of a 10kg mass:
a)100m above the surface of the earth
b)at the bottom of a vertical mine shaft 1000m deep.
Sol.
3.5.2 Types of Potential Energy (h)
Based on the position of the body, we can say that the potential energies basically are of following types:
1. Gravitational potential energy 2. Electric potential energy 3. Elastic potential energy 4. Electromagnetic potential energy 5. Nuclear potential energy 6. Chemical Potential energy
3.5.2.1 Gravitational potential energy (sh)
It is defined as the energy acquired by an object by virtue of its height above earth.When an object is taken to a certain height, the work done on it is the product of the force due to gravity and the height. Hence the Gravitational potential energy of an object is defined mathematically as,
P = mgh ........................(1) where, P is the potential energy of an object of mass m at a height h from the ground g is the acceleration due to gravity at that place The above relation is called as Gravitational Potential Energy Formula. Since the acceleration due to gravity anywhere is generally considered as a constant,
1. For a given mass the potential energy of an object is directly proportional to its height above the ground and
2. For a given height above the ground the potential energy of an object is directly proportional to its mass.
3.5.2.2 Electric potential energy (sh)
Some objects can be electrically charged due to electron movements on the surface under some conditions. Such objects under charged condition create an electromagnetic field which tends to oppose any other electrical charge and this ability to oppose is called Electric Potential
Energy of the initial charge.Now to bring another electrical charge to this initial charge, field work has to be done against the force of the same field.
Let us assume that an object is charged to an unit of Q. Suppose another charge of magnitude q is present at a distance r, the force F exerted by the Q on q is given by Coulomb’s law which states, F = k ×Q×qr2
where, k is a constant,
F is the force exerted by the charge Q on q,
r is the distance between the two charges.
Now the work done dw against the electromagnetic field to move the charge towards by a distance dr
dW = -Fdr = (−k×Q×q)/r2 dr, The negative sign is due to the fact that the force is opposing. Unit of Work is joule (J). Hence the total work done from infinity to a distance r is, W = ∫r∞dW = (k×Q ×q)/r
This is nothing but the Electric Potential Energy of the charge Q.
3.5.2.3 Elastic Potential Energy (sh)
An elastic object is said to possess an elastic potential energy when the object is subjected to a tension or compression by an external force. The elastic potential energy of a spring is the most common example.
A Spring is stretched or compressed by applying a force and the amount of stretch or compression is called the displacement. The force required to do this work is given by Hooke’s law as, F = kd where, k is a constant known as Spring Constant and d is the length of Displacement of the end of the spring However, the average force requires for the displacement is F/2. Hence the work done for the displacement is F/2 × d = kd2, which is stored as the elastic potential energy of the Spring.
3.5.2.4 Nuclear Potential Energy (sh)
Nuclear force is the Short range force existing in the universe. We also know that nucleus is made up of protons and neutrons that binds the nucleus together.
Thus, nuclear potential energy can be defined as:
The sum of potential energy of all the protons and neutrons in a nucleus due to the nuclear forces between them, excluding the electrostatic potential energy existing between them is called nuclear potential energy.
3.5.2.5 Chemical Potential Energy (sh)
The Chemical potential energy is the energy which is recognized when an object releases or absorbs energy if it undergoes a Chemical reaction.Generally, it can be classified as:
1. Exothermic Reaction: In this type of reaction, heat is released 2. Endothermic Reaction: In this type of reaction, heat is absorbed
Example: When a coal is burnt, it undergoes a chemical reaction with oxygen. Energy is released in that process in the form of heat. The amount of heat that can be released per unit weight is called as calorific value and it is the measure of potential energy.
3.5.2.6 Potential Energy Diagram (sh)
The Change in potential energy of an object occurs normally for an instantaneous time except in some cases of gravitational potential energy and chemical potential energy. If we plot a graph for these Variations of potential energy with respect to time, we call it as Potential Energy Diagram. or The graph of variation of potential energy with respect to instantaneous time is called Potential Energy Diagram. Potential energy diagrams are more prominent in case of chemical reactions and important for certain studies. Let us consider this potential energy diagram:
3.5.3 Examples of Potential Energy (h)
Following are the applications of potential energy:
1. Hydro Electric Power Plant: Water is stored in a dam at a considerable height and a particular quantum is allowed gravity flow to create a kinetic energy. This kinetic energy is converted as rotational energy of an electric generator by proper coupling. Finally, the Rotational energy is converted into electrical energy.
2. Nuclear Power Plant: The Nuclear energy released from nuclear potential energies are used both for productive and destructive uses. Generation of power is a positive use. This concept is used in nuclear power plants throughout the world. On the other hand use of nuclear potential energy as a weapon like nuclear bombs.
3. Spring in the Form of Elastic Potential Energy: The Springs on a vehicle act as cushion absorbing the shocks they are subjected due to road conditions. The force of
shocks are absorbed by the springs in the form of elastic potential energy. Electric potential energy is also used as the source of power in electric motors.
4. To Get Electric Power: The Chemical potential energy of fuels is used in generation of heat in boilers which in turn is used in generation of electric power.
Example- A body of mass 2kg is raised to a height of 25m. Calculate the potential energy possessed by the body?
Solution:
Mass of the body (m) = 2kg, Height (h) = 25 m, Potential Energy possessed by the body = mgh g = 9.8 m/s2
Potential Energy = 2 kg × 9.8 m/s2 × 25 m = 50 kg m × 9.8 m/s2 = 490 Nm = 490 J Example- A bag of rice weighs 75kg. To what height is it raised if the work done in lifting it is 4900 J? Solution:
Mass of the bag of rice = 75 kg, Work done = 4900 J, Work done in lifting the bag of rice is stored in it as its potential energy.
Potential energy = mgh 4900 J = 75 kg × 9.8 m/s2 × h Height, h = 4900J75Kg×9.8ms2 = 4900J735Kgm/s2 = 6.67 m.
3.6 Kinetic Energy (mh)
Kinetic energy may be described as energy due to motion.The kinetic energy of a body may be defined as the amount of work it can do before being brought to rest.
For example when a hammer is used to knock in a nail, work is done on the nail by the hammer and hence the hammer must have possessed energy.Only linear motion will be considered here.
3.6.1 Formulae for kinetic energy (h)
Let a body of mass m moving with speed v be brought to rest with uniform deceleration by a constant force F over a distance s.
And work done is given by
The force is F = ma so
Thus the kinetic energy is given by
3.6.2 Kinetic energy unit (h)
Dimensionally, the kinetic energy of any object is equivalent to the product of its mass and the square of its velocity.
In fps system of units, the unit of kinetic energy is lbm-(ft/s)2.
In SI system of units, the unit of kinetic energy becomes as kgm2/s2 better known as Joule.
3.6.3 Types of Kinetic energy (h)
There are basically four types of Kinetic Energy based on its motion:
1. Translational Kinetic energy 2. Turbulent Kinetic energy
3. Relativistic kinetic energy 4. Negative kinetic energy
The Energy possessed by the body when the body is moving along the straight line is called Translational Kinetic energy.
Turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow.
Relativistic Kinetic energy is the Kinetic energy possessed by the body acquiring velocity comparable to the velocity of light.
Negative kinetic energy is the kinetic energy possessed by the body when its velocity decreases as compared to its initial Velocity.
3.6.4 Average Kinetic energy (h)
By default kinetic energy means only an average kinetic energy. This is because the distance for the work done is calculated on the average of final and initial velocities in a time interval. Hence, for the translational kinetic energy (meaning the kinetic energy of a linear motion), the average kinetic energy is defined as,
E = 1/2 mv2
or average Velocity v = (V+U)/t,
where U = initial velocity,
V = final velocity,
t = time taken.
The velocity is considered at the center of mass.
3.6.5. Potential and kinetic energy (h)
The most common set in transformation of energies is potential and kinetic energy. A potential energy is defined to be an energy by virtue of its position of an object. Suppose an object of mass ‘m’ is at a height of ‘h’, its potential energy P is defined as: P = mgh, where m is the mass, g is the acceleration due to gravity
h is the height from ground. Suppose the same object is dropped from that height, it reaches the ground with a velocity ’v’, thus acquiring a kinetic energy
E = 1/2 mv2 At this point the entire potential energy is dissipated and as per conservation of energy, P.E = K.E or P = E
mgh = 1/2 mv2
or h = v2/2g
Which gives an important relation between the height and the final velocity of falling objects.
3.6.6 Relativistic Kinetic energy (h)
In our discussion of kinetic energy, we assumed velocity of an object that can be negligible when compared to the velocity of light. The study with such assumption is called classical mechanics. But after the introduction of theory of relativity, a review becomes necessary when the velocity of objects in motion is comparable with velocity of light. In such a case the mechanics is called Relativistic Mechanics.
In relativistic mechanics, the kinetic energy is upgraded as Relativistic Kinetic Energy.
The relativistic kinetic energy of a rigid object of mass m with actual velocity v is defined as, E = mc2(k-1)
where, c is the velocity of light and
k is given by the relation k2 = 1/(1–(v/c)2)
3.6.7 Examples of kinetic energy (h)
Example- A truck and a car are moving with the same kinetic energy on a road. Their engines are simultaneously switched off. Which one will stop at a lesser distance? Solution: The vehicle stops when its kinetic energy is spent in working against the force of friction between the tyres and the road. This force of friction varies directly with the weight of the vehicle. As the K.E. = work done = Force of friction × distance E = F × d or
d = E/F. where E = Kinetic energy, F = Force For given kinetic energy, distance s will be smaller, where F is larger, such as in case of truck. Thus truck stops Earlier. Example- A body of mass 75 kg has a momentum of 1500kg ms-1. Calculate its kinetic energy? Solution: As momentum p = m × v. where m = Mass = 75 kg, v = Velocity p = 1500 kg ms-1 1500 kg ms-1 = 75 kg × v v = 20m/s. Kinetic energy, K.E. = 12 mv2 = 12 × 75kg × 400 m2/s2 = 15000 Joules. Example- Given that the displacement of body in meters is a function of time as follows: x = 2t4 + 5. The mass of the body is 2kg. What is the increase in its kinetic energy one second after the start of the motion?
Solution: Velocity = v = dxdt = ddt (2t4 + 5) = 8t3 After one second velocity is 8(1)3 = 8 m/s Kinetic energy, K = 12 mv2 where m = Mass of the body, v = Velocity K = 12 × 2 kg × (8)2 m2/s2 = 64J. Example- If we throw a body upwards with velocity of 4ms-1, then at what height its kinetic energy reduces to half of the initial value? Take g = 10 ms-2.
Solution: Initial energy = 12 m(4)2 = 8m m/s2. Let kinetic energy at a height h be 82 = 4m m/s2. It will also be equal to the potential energy. Hence K.E. = P.E 12 mv2 = mgh. 4m m/s2 = m × 10 m/s2 × h h = 0.4m.
3.6.8 Kinetic energy and work done (h)
When a body with mass m has its speed increased from u to v in a distance s by a constant force F which produces an acceleration a, then
multiplying this by m give an expression of the increase in kinetic energy (the difference in kinetic energy at the end and the start).Thus since F = ma
but also we know
So the relationship between kinetic energy can be summed up as
Work done by forces acting on a body = change of kinetic energy in the body.
This is sometimes known as the work-energy theorem.
Example- A car of mass 1000 kg travelling at 30m/s has its speed reduced to 10m/s by a constant breaking force over a distance of 75m.Find:
1.The cars initial kinetic energy
2.The final kinetic energy
3.The breaking force
Solution.
Change in kinetic energy = 400 kJ
3.7 Conservation Of Energy (mh)
The principle of conservation of energy state that the total energy of a system remains constant. Energy cannot be created or destroyed but may be converted from one form to another.
Take the case of a crate on a slope. Initially it is at rest, all its energy is potential energy. As it accelerates, some of it potential energy is converted into kinetic energy and some used to overcome friction. This energy used to overcome friction is not lost but converted into heat. At the bottom of the slope the energy will be purely kinetic (assuming the datum for potential energy is the bottom of the slope.)
If we consider a body falling freely in air, neglecting air resistance, then mechanical energy is conserved, as potential energy is lost and equal amount of kinetic energy is gained as speed increases.
If the motion involves friction or collisions then the principle of conservation of energy is true, but conservation of mechanical energy is not applicable as some energy is converted to heat and perhaps sound
Example- A cyclist and his bicycle has a mass of 80 kg. After 100m he reaches the top of a hill, with slope 1 in 20 measured along the slope, at a speed of 2 m/s. He then free wheels the 100m to the bottom of the hill where his speed has increased to 9m/s.How much energy has he lost on the hill?
Sol.
If the hill is 100m long then the height is:
So potential energy lost is
Increase in kinetic energy is
By the principle of conservation of energy
.
3.8 Review Questions (mh)
1. A 1200 kg car and a 2400 kg car are lifted to the same height at a constant speed in a auto service station. Lifting the more massive car requires what amount of work? Presume that the value of g is ~10 m/s2.
2. A child lifts a box up from the floor. The child then carries the box with constant speed to the other side of the room and puts the box down. How much work does he do on the box while walking across the floor at constant speed? Presume that the value of g is ~10 m/s2.
3. Which requires more work: lifting a 50.0 kg crate a vertical distance of 2.0 meters or lifting a 25.0 kg crate a vertical distance of 4.0 meters? Presume that the value of g is ~10 m/s2.
4. What will be the kinetic energy of an arrow having potential energy of 50 J after it is shot from a bow?
5. If 8 million kg of water flows over Niagara Falls each second, calculate the power available at the bottom of the falls.
6. A lever is used to lift a heavy load. When a 50-n force pushes one end of the lever down 1.2 m, the load rises 0.2 m. Calculate the weight of the load.
7. What is the efficiency of her body when a cyclist expends 1000 W of power to deliver mechanical energy to the bicycle at the rate of 100 W?
CHAPTER-04
ROTATIONAL AND SIMPLE HARMONIC MOTION (ch)
4.1 Rotational Dynamics (mh)
As we have linear motion, similarly we have rotational motion. We have laws of motion for linear motion in the same way as we have it for rotational motion. Dynamics includes the study of the effect of torques in motion. So, Rotational Dynamics is all about study of effect of torque acting on the rotating body.
4.1.1 Angular Acceleration (h)
To understand the Rotational dynamics, we need to know about many terms like torque, angular acceleration etc. Angular displacement is denoted by θ. For a particle moving in a circle of radius r and assuming that it has moved an arc length of s, the angular position theta is given by, θ = s/r
Unit: radians
1 radian = 180/π
Angular Displacement: The angular displacement can be defined as the change in the angular position of the particle or object.
Δ θ = θ2 - θ1 θ2 = Final angular position
θ1 = Initial angular position
Angular Velocity and Angular Acceleration
The rate of change of angular displacement is called Angular velocity.
Unit: radian per second
ω¯¯ = ΔΘ/Δt ω = 2πf
where, ω = angular velocity
Angular acceleration: The rate at which angular velocity changes with time is called Angular acceleration. A¯¯¯ = Δω/Δt Here A is pronounced as alpha
Also a = rα
where, a = translational acceleration
4.1.2 Rotational Dynamics Equations
There are 3 equations. They are,
ω(t) = ω0+ at q(t) = q0 + ω0t + 1/2at2 ω0 = ω2
0 + 2a (q - q0) where, ω0 = magnitude of the initial angular velocity ωt = angular velocity’s magnitude after time t q0 = Initial angular position q(t) = Angular position after time t
4.1.3 General Rolling Motion (h)
General rolling motion consists of both translation and rotation. Although analysis of the general rotary motion of a rigid body in space may be quite complicated, it is made easier by a few simplifying constraints. Initially we will consider only objects with an extremely high degree of symmetry about a rotational axis, e.g., hoops, cylinders, spheres.Consider a uniform cylinder of radius R rolling on a rough (no slipping) horizontal surface.
As the cylinder rotates through an angular displacement θ, its center of mass (cm) moves through distance s = rθ , or the same distance as the arc length.
scm = Rθ
vcm=∆s/∆t=R∆θ/∆t=ωR
acm=∆vcm/∆t=R∆ω/∆t=Rα
4.2 Definition of Moment of Inertia (mh)
Moment of inertia is a property of a body that defines its resistance to a change in angular velocity about an axis of rotation. It is how rotation of a body is affected by Newton's law of inertia, which states "Every body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed."In this context, inertia refers to resistance to change.Moment of inertia applies to an extended body in which the mass is constrained to rotate around an axis. It arises as a combination of mass and geometry in the study of the movement of continuous bodies, or assemblies of particles, known as rigid body dynamics. It is the moment of inertia of the pole carried by a tight-rope walker that resists rotation and helps the walker maintain balance.
4.2.1 Calculating moment of inertia (h)
Consider the kinetic energy of an assembly of N masses mi that lie at the distances ri from the pivot point P, which is the sum of the kinetic energy of the individual masses
This shows that the moment of inertia of the body is the sum of each of the mr2 terms, that is, thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
4.2.2 Example calculation of moment of inertia (h)
• The moment of inertia of a thin rod with constant cross-section s and density ρ and with length l about a perpendicular axis through its center of mass is determined by integration.Align the x-axis with the rod and locate the origin its center of mass at the center of the rod, then
where m = πR2ρs is its mass.
• Moment of inertia of sphere- consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moment of inertias of the thin discs that form the sphere. If the surface of the ball is defined by the equation.
then the radius r of the disc at the cross-section z along the z-axis is
Therefore, the moment of inertia of the ball is the sum of the moment of inertias of the discs along the z-axis,
where m = (4/3)πR3ρ is the mass of the ball.
4.3 Angular momentum in planar movement (mh)
The angular momentum vector for the planar movement of a rigid system of particles is given
Use the center of mass C as the reference point so
and define the moment of inertia relative to the center of mass IC as
then the equation for angular momentum simplifies to
4.3.1 Definition and relation of torque to angular momentum (h)
A force applied at a right angle to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm) is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. The direction of the torque can be determined by using the right hand grip rule: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque.More generally, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:
where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude τ of the torque is given by
where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and θ is the angle between the position and force vectors. Alternatively,
where F is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque.It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. It points along the axis of the rotation that this torque would initiate, starting from rest, and its direction is determined by the right-hand rule.The unbalanced torque on a body along axis of rotation determines the rate of change of the body's angular momentum,
where L is the angular momentum vector and t is time. If multiple torques are acting on the body, it is instead the net torque which determines the rate of change of the angular momentum:
For rotation about a fixed axis,
where I is the moment of inertia and ω is the angular velocity. It follows that
where α is the angular acceleration of the body, measured in rad/s2. This equation has the limitation that the torque equation is to be only written about instantaneous axis of rotation or center of mass for any type of motion - either motion is pure translation, pure rotation or mixed motion. I = Moment of inertia about point about which torque is written (either about instantaneous axis of rotation or center of mass only). If body is in translatory equilibrium then the torque equation is same about all points in the plane of motion.
4.4 Principle of Conservation of Energy (mh)
If a compound body swings, under gravity, we can find its angular speed at a subsequent instant,
by invoking the Principle of conservation of energy: is a constant
4.5 Rotational Energy (mh)
The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, one gets the following dependence on the object's moment of inertia:
Where w is the angular velocity
I is the moment of inertia around the axis of rotation
E is the kinetic energy
The mechanical work required for / applied during rotation is the torque times the rotation angle. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. For free-floating (unattached) objects, the axis of rotation is commonly around its center of mass
in the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, \omega , takes the role of the linear velocity, v. The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow).
Example- Determine the radius of gyration ky of the parabolic area.
A. ky = 76.5 mm
B. ky = 17.89 mm
C. ky = 78.6 mm
D. ky = 28.3 mm
Sol. B
Example- Determine the inertia of the parabolic area about the x axis.
A. Ix = 11,430 in.4
B. Ix = 32,800 in.4
C. Ix = 13,330 in.4
D. Ix = 21,300 in.4
Sol. A
Example- Calculate the Moment of inertia of the ball having mass of 5 Kg and radius of 3 cm?
Sol. Given: Mass of the ball = 5Kg,
Radius of the ball = 3 cm = 0.03 m,
Moment of Inertia is given by I = MR2
= 5 Kg × (0.03 m)2
= 0.0045 Kgm2.
Example- A sphere is moving around in air. If the moment of inertia is 10 Kgm2 and radius of 1m, Calculate its mass?
Sol: Moment of inertia I = 10 Kgm2,
Radius of sphere R = 1m,
Moment of Inertia I = MR2
Mass of the body M = I/R2
= 10Kgm2/1
= 10 Kg.
4.6 Simple Harmonic Motion (mh)
In mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. It can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is
sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement.
4.6.1 Dynamics of simple harmonic motion (h)
For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.
where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is the spring constant.Therefore,
Solving the differential equation above, a solution which is a sinusoidal function is obtained.
where
In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the phase.
Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:
Acceleration can also be expressed as a function of displacement:
Then since ω = 2πf,
and since T = 1/f where T is the time period,
These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).
Example- A mass of 2 kg is attached to a spring with constant 18 N/m. It is then displaced to the point x = 2 . How much time does it take for the block to travel to the point x = 1 ?
Sol. For this problem we use the sin and cosine equations we derived for simple harmonic motion. Recall that x = x m cos(σt) . We are given x and x m in the question, and must calculate
σ before we can find t . We know, however, that no matter the initial displacement, σ = =
= =3
=cosσt
=cos3t
3t = cos-1 (1/2)
t = = .35 seconds
Example- A mass of 2 kg oscillating on a spring with constant 4 N/m passes through its equilibrium point with a velocity of 8 m/s. What is the energy of the system at this point? From your answer derive the maximum displacement, x m of the mass.
Sol. When the mass is at its equilibrium point, no potential energy is stored in the spring. Thus all of the energy of the system is kinetic, and can be calculated easily:
K = mv 2 = (2)(8)2 = 64 Joules
Since this is the total energy of the system, we can use this answer to calculate the maximum displacement of the mass. When the block is maximally displaced, it is at rest and all of the energy of the system is stored as potential energy in the spring, given by U = kx m2 . Since energy is conserved in the system, we can relate the answer we got for the energy at one position with the energy at another
E f = E o
kx m2 = mv 2= 64.
4.7 Review Questions (mh)
1. A hollow sphere is of mass M, external radius a and internal radius xa. Its rotational inertia is 0.5 Ma2. Show that x is given by the solution of
1 − 5x 3 + 4x 5 = 0
and calculate x to four significant figures. (Answer = 0.6836.)
2. What is the direction of the angular velocity vector for the second hand of a clock going from 0 to 30 seconds?
3. What is the torque on the pivot of a pendulum of length R and mass m, when the mass is at an angle ?
4. A disk of mass m and radius R rolls down an inclined plane of height h without slipping. What is the velocity of the disk at the bottom of the incline? The moment of inertia for a disk is 1 /2 mR2.
5. How should the mass of a rotating body of radius r be distributed so as to maximize its angular velocity?
6. A catapult with a basket of mass 50 kg launches a 200 kg rock by swinging around from a horizontal to a vertical position with an angular velocity of 2.0 rad/s. Assuming the rest of the
catapult is massless and the catapult arm is 10 m long, what is the velocity of the rock as it leaves the catapult?
7. Two objects rest on a seesaw. The first object has a mass of 3 kg and rests 10 m from the pivot. The other rests 1 m from the pivot. What is the mass of the second object if the seesaw is in equilibrium?
8. What is the velocity of a point on the rim of the standard 12-inch long-playing phonograph record?
R = 15.2 cm
ω = 33 (1/3) rpm = 5.56 x 10-1 rev/sec = 3.49 rad/sec.
v = ωR = 53.1 cm/sec
9. A butcher throws a cut of beef on spring scales which oscillates about the equilibrium position with a period of T = 0.500 s. The amplitude of the vibration is A = 2.00 cm (pathlength 4.00 cm). Find:
a. frequency
b. the maximum acceleration
c. the maximum velocity
d. the acceleration when the displacement is 1.00 cm
e. the velocity when the displacement is 1.00 cm
f. the equation of motion as a function of time if the displacement is A at t = 0.
10. It is known that a load with a mass of 200 g will stretch a spring 10.0 cm. The spring is then stretched an additional 5.00 cm and released.
Find: a. the spring constant
b. the period of vibration and frequency
c. the maximum acceleration
d. the velocity through equilibrium positions
e. the equation of motion
CHAPTER-05
TEMPERATURE AND ITS MEASUREMENT (ch)
5.1 Temperature (mh)
Temperature of a substance is defined as the measure of the internal energy contained in the substance. Temperature is directly related to the kinetic energy of the particles, atoms and molecules of the substance. It is the average kinetic energy of the particles contained in the system.
It determines the direction of transfer of heat when two substances at different temperatures come in contact. In simple terms it can also be stated as the degree of hotness or coldness of the system.
Temperature is an intensive property: that is it does not depend on the amount of substance. Thus one kilogram of copper at 80 °C and 12 kg of copper at 80 °C both have the same temperature. Note that unless we are dealing with radiated heat, it is not normally necessary to change these values to the Absolute Temperature scale. The Celsius temperature is simply defined as the number of kelvin above 273.15 K. If we wish to calculate heat transfer from these blocks of copper to water at 20 °C, it is quite adequate to say the temperature difference is 80 °C - 20 °C = 60 K. We get the same answer with more effort by saying it is 353
– 293 = 60 K. (As I am working to the nearest degree, I have omitted the 0.15 K). Temperatures may be given on the Absolute or Celsius temperature scales, but temperature differences should be given in kelvin.
5.2 Units of Heat Energy (mh)
Heat being a form of energy can be measured. What is the unit of heat energy?
In CGS system, heat is measured in calories. A calorie is defined as 'the heat energy required to increase the temperature of 1gm of water through one degree Celsius'.
In SI system, heat energy is measured in joules (J). Infact, in SI system all forms of energy is measured in joules.
One calorie = 4.185 joules.
A larger unit called Kilocalorie is also in use. 1 kilocalorie = 1000 calories.
Energy contents of food is usually measured in kilocalories. The calories counted in food are actually kilocalories. For example, the energy value of 100 g of baked potato is equal to 100 calories or 100 kilocalories..
Example- Calculate the Heat lost by the block when iron block decreases its temperature from 60oC to 40oC if the mass of the body is 2 Kg. Specific heat of iron C = 0.45 kJ/kg K.
Sol: Given: Initial temperature Ti = 600C,
Final temperature Tf = 400C,
Mass of the body m = 2 kg,
The Heat lost is given by Q = m c Δ T
= 2 Kg × 0.45 kJ/kg K × 293 K
= 263.7 J.
5.3 Temperature Scales (mh)
Much of the world uses the Celsius scale (°C) for most temperature measurements. It has the same incremental scaling as the Kelvin scale used by scientists, but fixes its null point, at 0°C = 273.15K, approximately the freezing point of water (at one atmosphere of pressure). The United
States uses the Fahrenheit scale for common purposes, a scale on which water freezes at 32 °F and boils at 212 °F (at one atmosphere of pressure).
For practical purposes of scientific temperature measurement, the International System of Units (SI) defines a scale and unit for the thermodynamic temperature by using the easily reproducible temperature of the triple point of water as a second reference point. The reason for this choice is that, unlike the freezing and boiling point temperatures, the temperature at the triple point is independent of pressure (since the triple point is a fixed point on a two-dimensional plot of pressure vs. temperature). For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment, which has been named the kelvin in honor of the Scottish physicist who first defined the scale. The unit symbol of the kelvin is K.
Absolute zero is defined as a temperature of precisely 0 kelvins, which is equal to −273.15 °C or −459.67 °F.
5.4 Temperature Conversion
Mainly used temperature units are Celsius(°C), Fahrenheit(°F) and Kelvin (K). Celsius(°C), Fahrenheit(°F) are relative units of the temperature, whereas Kelvin is the absolute unit of temperature.
1. Celsius(°C) – Earlier it was known as centigrade scale. 0°C is termed as the freezing point of the water and 100°C is the boiling point of water under 1 atmospheric pressure unit.
Temperature Conversion Formula
To convert celsius values to Fahrenheit or Kelvin values and vice versa we use the
From Celsius(°C) To Celsius(°C)
Fahrenheit [°F] = [°C] × 95 + 32 [°C] = ([°F] - 32) x 59
Kelvin [K] = [°C] + 273.15 [°C] = [K] - 273.
2. Fahrenheit ( F) - In F scale, freezing point of water is 32 F and the boiling point is 212 °F, and the boiling and freezing points of water are 180 degrees apart. A single °F is 1180th part of interval between the ice point and boiling point of water. A temperature interval of one degree Fahrenheit is an interval of 59 of a degree Celsius. Both the scales (Fahrenheit and Celsius) coincide at −40 degrees (i.e. −40 °F = −40 °C). Absolute zero in Fahrenheit is −459.67 °F. Conversion Formulae
To convert Fahrenheit values to Celsius or Kelvin values and vice versa we use the formula –
From Fahrenheit To Fahrenheit
Celsius [oC] = ([oF]) -32) × 59 [oF] = [oC] × 95 + 32
Kelvin [K] = ([oF] + 459.67) × 59 [oF] = [K] × 95 - 459.67
3. Kelvin (K)- Kelvin is the absolute temperature unit. One Kelvin can be explained as the 1/273.16 of the temperature of triple point of water – (i.e. 273.16 K). Kelvin is the SI temperature unit. Absolute temperature is the temperature at which the energy of the substance becomes zero.
To convert Kelvin values to Celsius or Fahrenheit values and vice versa we use the formula – Conversion Formulae:
From kelvin To Kelvin
Celsius [oC] = [K] - 273.15 [k] = [oC] + 273.15
Fahrenheit [oF] = [K] X 95 - 459.67 [K] = ([oF] + 459.67 ) × 59
Example- Convert 50 °F to Celsius scale. Solution: Now here we have temperature in Fahrenheit scale. We need to convert it into the Celsius scale. Formula to convert temperature in Fahrenheit to Celsius is °C = ( F−32)1.8 = (50–32)1.8 = (18)1.8 = 10 °C Example- Convert 10°C to °F. Solution:
Now here we have temperature in Celsius scale. We need to convert it into the Fahrenheit scale. Formula to convert temperature in Celsius to Fahrenheit is °F = 1.8 × °C + 32 =1.8 × 10 + 32 = 18 + 32
= 50 °F Example- Convert 10 °C to K. Solution: Now here we have temperature in Celsius scale. We need to convert it into the Kelvin scale. K = °C + 273.15 = 10 + 273.15 = 283.15 K
Example- Convert 50 °F to K. Solution: Fahrenheit to Kelvin Conversion Fahrenheit to Kelvin Formula: K = (°F) + 459.67 ) x 59 = (50 + 459.67) x 59 = 283.15
Example- Convert 30 K to Celsius scale. Solution: To convert K to Celsius scale Formula is °C = K - 273.15 = 30 - 273.15 = -243 °C Example- Convert 30 K to Fahrenheit scale. Solution: Kelvin to Fahrenheit Conversion: Kelvin to Fahrenheit Formula = K × 95 - 459.67 = 30 × 95 - 459.67 = 69 - 459.67 = 54 - 459.67 = - 405.67
5.4.1 Change of Phase/State (Phase Transition) (h)
Change of Phase: Matters can be in four states like solid, liquid, gas and plasma. Distance between the molecules or atoms of the matter shows its state or phase . Temperature and pressure are the only factors that affect the phases of matter. Under constant pressure, when you heat
matter, its speed of motion increases and as a result the distance between the atoms or molecules becomes larger. If you give heat to a solid substance, its temperature increases up to a specific point and after this point temperature of it is constant and it starts to change its phase from solid to liquid. Another example that all you in experience daily life, when you heat water it boils and if you continue to give heat it starts to evaporate. In this section we will learn these changes in the phases of substances and learn how to calculate necessary heat to change the states of them.
5.4.2 Melting and Freezing (h)
If solid matters gain enough heat they change state solid to liquid. Heat is a form of energy and in this situation it is used for the break the bonds of the atoms and molecules. Heated atoms and molecules vibrate more quickly and break their bonds. We call this process melting changing state solid to liquid. Inverse of melting is called freezing, changing state liquid to solid, in which atoms and molecules lost heat and come together, their motion slows down and distance between them decreases.
This is a phase of change of water from solid to liquid. As you can see at the beginning ice is at -15 ºC, we give heat and its temperature becomes 0 ºC which is the melting point of ice. During melting process temperature of the ice-water mixture does not change. After all the mass of ice is melted its temperature starts to rise.
Every solid matter has its own melting point; we can say that melting point is a distinguishing property of solids. The Inverse of this process is called freezing in which liquid lost heat and change phase liquid to solid. Freezing point and melting point are the same for same matter and it is also distinguishes property of matter.
5.5 Thermodynamics (mh)
Thermodynamics is a macroscopic science which studies various interactions amongst energy, notably heat and work transfer, with matter that brings about significant changes in the macroscopic properties of a substance that are measurable. It is basically a phenomenological science based on certain laws of nature which are always obeyed and never seen to be violated. These are the basic laws of thermodynamics.
Thermodynamics deals with the equilibrium, energy and its transformation from one form to another. It deals with the relationships between heat and work, and the properties of the system in equilibrium. Historically, Thermodynamics originated as a result of man's endeavor to convert heat into work.
The principles of thermodynamics are summarized in the form of four laws known as zeroth, first, second, and third law of thermodynamics.
It does not deal with the microscopic constituents of the matter. It deals with various macroscopic variables like pressure, temperature etc. The results of thermodynamics are useful for other branches and fields of physics and engineering like mechanical, chemical, physical, biomedical etc.
5.5.1 Law of Thermodynamics (h)
Thermodynamics is basically a phenomenological science based on certain laws of nature which are always obeyed and never seen to be violated. These are the basic laws of thermodynamics. These are the laws:
1. Zeroth law of thermodynamics
2. First law of thermodynamics
3. Second law of thermodynamics
4. Third law of thermodynamics
5.5.1.1 Zeroth Law of Thermodynamics (sh)
The basic property which distinguishes thermodynamics from other sciences is temperature. Temperature is as important to thermodynamics as force is to statics. Temperature is basically associated with the ability to distinguish hot from cold. When two bodies having different temperatures come in contact with each other, they attain a common temperature and hence the state of thermal equilibrium.
According to the Zeroth law of thermodynamics: When a given object or body A is in thermal equilibrium with a body or object B, and separately with a body or object C also, then B and C would necessarily be in thermal equilibrium with each other.
The Zeroth law of thermodynamics is the basis for temperature measurement of a body or object. A body at a lower temperature is called a cold body and a body at a higher temperature is called a hot body.
Thermodynamic system follows these properties:
1. Symmetric: If a body K is in thermal equilibrium with a body L then it is necessary that the opposite would also be true, i.e., L would be in thermal equilibrium with K.
2. Euclidean relations between bodies.
3. Reflexive relation between bodies.
4. Equivalence relation between bodies.
5. Transitive relation between bodies that is if K is in thermal equilibrium with L and L is in thermal equilibrium with M then it is necessary that K will be in thermal equilibrium with M.
Thermometric properties and thermometer: A physical characteristic of an arbitrarily chosen body which changes with change in temperature is called thermometric property (X). The reference body is then called the thermometer. There are 5 different kinds of thermometers each with its own thermometric property. The most common type of thermometer consists of a small amount of mercury in an evacuated capillary tube.
The common thermometers are mercury in glass, resistance, thermocouple, constant volume gas thermometer and constant pressure gas thermometer. Before 1954, two fixed points called the ice point and the steam point were used to measure the temperature of a system. After it only one point was used called the triple point of water which is the standard fixed point of thermometry. An absolute temperature scale may be constructed by taking the temperature directly proportional to the volume of the gas in a constant pressure gas thermometer. It is known
that the scale which is based upon a gas at zero pressure is the physical realization of a logically formulated absolute thermodynamics temperature scale, which will be introduced in connection to second law of thermodynamics. For ordinary purpose absolute temperature may be found from the relation ( C = 273.15 K). 5.5.1.1.1 Zeroth Law of Thermodynamics Examples (ssh)
• Let us consider two beakers full of water. Then for one beaker, the temperature of water is above the normal room temperature, and for the other beaker it is below the normal room temperature. They are left on the table for some time such that they both are not in contact with each other. If we check the beakers after some time, equilibrium for both the beakers is reached. As observed both the beakers of water are at the same temperature. The two beakers actually come in thermal equilibrium with the surroundings. Hence they are in thermal equilibrium with each other also, and they are at the same temperature.
• When we take an electric rod and put it in water then the water also becomes hot. This is because the heat is exchanged between the two in order to come in thermal equilibrium with each other.
• Sweating in human body is another example. We feel a cooling effect after sweating.
5.5.1.2 First Law of Thermodynamics (sh)
If some amount of heat is given to the closed system, a part of heat is used to increase internal energy of the system and remaining part is used as the work done by the system.This statement was stated by Rudolf Clausius as the First law of thermodynamics in 1850. Mathematically, it can be stated as:
Δ Q = Δ U + Δ W Where, Δ Q = Heat given to the system, Δ U = Increase in internal energy of the system Δ W = Work done by the system. Thus, the change of internal energy of the system is given by:
Δ U = Δ Q - Δ W Thus, we can also state it as: “The change in the internal energy of the system is equal to the amount of heat supplied to the system minus the amount of work performed by the system on its surroundings”. Hence the first law of thermodynamics can be stated as: “The energy of the system remains constant given the system is isolated”. The first law of thermodynamics was also stated using the cyclic process by Clausius as:
∑ Q cycle = ∑ Wcycle
where, ∑ Qcycle = Heat given to the cyclic process
∑ Wcycle= Work done in the cyclic process
5.5.1.2.1 First Law of Thermodynamics Example (ssh)
1) Consider gas in the cylinder fitted with piston.
If the cylinder is heated, increase in temperature takes place, the energy of gas molecules increase which leads to increase in internal energy of the system.
or if the gas is compressed using the piston it also increases the internal energy of the system.
2) The evaporation of sweat from human body is a perfect example of the first law of thermodynamics. Energy conservation is a fundamental law and hence works for many instances of our daily life.
5.5.1.3 Second Law of Thermodynamics (sh)
It is impossible for any system to undergo a process in which it absorbs heat from a reservoir at a single temperature and converts the heat completely into mechanical work with the system ending in the same state in which it began. We call this the 'engine' statement of the second law.
The basis of the second law of thermodynamics lies in the difference between the nature of internal energy and that of macroscopic mechanical energy. In a moving body the molecules have random motion but superimposed on this is a co-ordinated motion of every molecule in the direction of the body's velocity. The kinetic energy associated with this coordinated macroscopic motion is called the kinetic energy of the moving body. The kinetic energy and potential energy associated with the random motion constitute the internal energy.
When a body sliding on a surface comes to rest due to friction, the organized motion of the body is converted to random motion of molecules in the body and on the surface. Since we cannot control the motion of individual molecules, we cannot convert this random motion completely back to organized motion. Only a part of it can be converted and this is what a heat engine does.
If the second law was not true, we could have powered an automobile or run a power plant by cooling the surrounding air. Neither of these impossibilities violate the first law of thermodynamics. The second law, therefore, is not a deduction from the first, but stands by itself as a separate law of nature. The first law denies the possibility of creating or destroying energy and the second law limits the availability of energy and the ways in which it can be used and converted.
An alternative statement of the second law of thermodynamics states that heat flows spontaneously from hotter to colder bodies but the reverse is never true. A refrigerator transfers heat from a colder to a hotter body but its operation requires an input of mechanical energy or work. Generalizing this observation, we state:
“It is impossible for any process to sole result in the transfer of heat from a cooler to a hotter body. We call this the 'refrigerator' statement of the second law”.
Though, it may not seem to be very closely related to the 'engine' statement, the two statements are equivalent. For example, if we could build a refrigerator which does not have an input of work, violating the 'refrigerator' statement of the second law, we could use it in conjunction with a heat engine, pumping the heat rejected by the engine, back to the hot reservoir to be reused. This composite machine shown below in the figure (b) would violate the 'engine'
statement of the second law because its net effect would be to take a net quantity of heat QH - |QC| from the hot reservoir and convert it completely to work W.
Alternatively, if we could make an engine with 100% thermal efficiency, violating the first statement, we could run it using heat from the hot reservoir and use the work output to drive a refrigerator that pumps heat from the cold reservoir to the hot reservoir as shown in the figure (b) below. This composite device would violate the "refrigerator"� statement because its net effect would be to take heat QC from the cold reservoir and deliver it to the hot reservoir without requiring any input of work. Thus, any device that violates one form of the second law can be used to make a device that violates the other form. If violations of the first form are impossible so are violations of the second!
The conversion of work to heat as in friction or viscous fluid flow and heat flow from hot to cold across a finite temperature gradient are irreversible processes. The 'engine' and 'refrigerator' statements of the second law state that these processes can be partially reversed. We could cite other examples. Gases always seep spontaneously through an opening from a region of
high pressure to a region of low pressure: gases and miscible liquids left to themselves, always tend to mix, not to unmix.
For a heat engine: QH - QC= W
Efficiency of a heat engine = 1 – QH/QC
The second law of thermodynamics is an expression of the inherent one-way aspect of these and many other irreversible processes. Energy conversion is an essential aspect of all plant and animal life and of human technology. Hence, the second law of thermodynamics is of utmost fundamental importance to the world we live in.
5.5.1.3.1 Second Law of Thermodynamics Example (ssh)
• Lets have a look at the movie ‘Titanic’ once again. What happened when the ship hits the iceberg? The steel pipes were ripped open. Now can you imagine it the other way round? The ship is healing as it comes out of water and starts floating. That is improbable.
• Energy always disperses for example the heat of the burning pan in the surrounding environment.
• When we need more energy we use it in a highly concatenated form. For example our body gets energy from carbohydrates that store energy in a small space.
• The second law just says that energy always gets diluted, moves from a highly concatenated form to less one. The usefulness of energy will decrease as it moves and makes things happen.
5.5.1.4 Third Law of Thermodynamics (sh)
We know that entropy is the rate of change of disorder occurring in a system. Walther Nernst introduced the concept of entropy in the third law of thermodynamics which states that:
“For a perfect crystal at the absolute zero temperature, the entropy would be exactly equal to zero”.
When only one minimum energy state is possessed by a perfect crystal the law would hold true. If we consider systems such as glasses which are not perfect crystal then a generalized form of 3rd law would be: “When the temperature approaches zero, the randomness or entropy of a system would approach
a constant value”. The Constant value of entropy is called Residual Entropy and it should be noted that it is not necessarily zero.
5.5.1.4.1 Third Law of Thermodynamics Equation (ssh) The third law can be expressed as: limT→0 S = 0 ...............................(a) Here S = entropy which is expressed as J s-1K-1. T = absolute temperature which is expressed in K. It can also be written as: T → 0 S → Smin .....................................(b) Hence we can say if the temperature approaches zero then the entropy approaches to its minimum value. Another application of third law of thermodynamics: S = 2.303 Cp log T ..........................................(c) Using this we can find the absolute entropy of any substance at a given temperature T. Here Cp is the heat capacity of the substance at a constant pressure.
Limitations of the law:
1. Even at 0 K Glassy solids have entropy which is greater than zero.
2. Solids that have mixtures of isotopes do not possess zero entropy at 0 K. For example: Solid chlorine does not have zero entropy at 0 K.
3. Crystals of CO, N2O, NO, H2O, etc., do not possess perfect order even at 0 K, thus their entropy is not equal to zero.
5.5.1.4.2 Third Law of Thermodynamics Example (ssh)
We can consider H2O if we want to understand the concept of the third law of thermodynamics. Water exists in three different states:
1. Gaseous state 2. Liquid state 3. Solid state
In Gaseous state
The entropy or randomness is very high. Here we are talking about the randomness in the motion of the molecules of which the water is made up of. They move with very high entropy. In Liquid state
Now the randomness is reduced. It is not as free as the gaseous state and hence we can say that the entropy of the molecules is reduced. This is because the movement between the molecules is reduced. In Solid state
In this state the moment between molecules is almost zero. The entropy approaches almost zero value. This is because the molecules are packed very tightly in the solid state and hence the randomness is very low. This is when it is cooled at very low temperature or at an absolute zero temperature. Now if it cooled further than all the motion between the molecules would stop. This is because these are no free spaces for the motion of the particles. And hence the entropy becomes almost zero.
5.6 Thermocouple (mh)
A thermocouple consists of two dissimilar conductors in contact, which produce a voltage when heated. The voltage produced is dependent on the difference of temperature of the junction to other parts of the circuit. Thermocouples are a widely used type of temperature sensor for measurement and control and can also be used to convert a temperature gradient into electricity. Commercial thermocouples are inexpensive,[ interchangeable, are supplied with standard connectors, and can measure a wide range of temperatures. In contrast to most other methods of temperature measurement, thermocouples are self powered and require no external form of excitation. The main limitation with thermocouples is accurate; system errors of less than one degree Celsius (°C) can be difficult to achieve.
Any junction of dissimilar metals will produce an electric potential related to temperature. Thermocouples for practical measurement of temperature are junctions of specific alloys which have a predictable and repeatable relationship between temperature and voltage. Different alloys are used for different temperature ranges. Properties such as resistance to corrosion may also be important when choosing a type of thermocouple. Where the measurement point is far from the measuring instrument, the intermediate connection can be made by extension wires which are less costly than the materials used to make the sensor. Thermocouples are usually standardized against a reference temperature of 0 degrees Celsius; practical instruments use electronic methods of cold-junction compensation to adjust for varying temperature at the instrument terminals. Electronic instruments can also compensate for the varying characteristics of the thermocouple, and so improve the precision and accuracy of measurements.
5.6.1 Thermoelectric effect (h)
Seebeck effect-The Seebeck effect is the conversion of temperature differences directly into electricity and is named after the Baltic German physicist Thomas Johann Seebeck, who, in 1821, discovered that a compass needle would be deflected by a closed loop formed by two different metals joined in two places, with a temperature difference between the junctions. This was because the metals responded differently to the temperature difference, creating a current loop and a magnetic field. Seebeck did not recognize there was an electric current involved, so he called the phenomenon the thermomagnetic effect.
Peltier effect-The Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors and is named for French physicist Jean Charles Athanase Peltier, who discovered it in 1834. When a current is made to flow through a junction between two conductors A and B, heat may be generated (or removed) at the junction. The Peltier heat generated at the junction per unit time, Q, is equal to
Q = (ΠA-ΠB)I
where ΠA , ΠB is the Peltier coefficient of conductor A (B), and I is the electric current (from A to B)
Thomson effect-In many materials, the Seebeck coefficient is not constant in temperature, and so a spatial gradient in temperature can result in a gradient in the Seebeck coefficient. If a current is driven through this gradient then a continuous version of the Peltier effect will occur. This Thomson effect was predicted and subsequently observed by Lord Kelvin in 1851. It describes the heating or cooling of a current-carrying conductor with a temperature gradient.
If a current density J is passed through a homogeneous conductor, the Thomson effect predicts a heat production rate \dot q per unit volume of:
Q=κ J .∆ T
where ∆T is the temperature gradient and κ is the Thomson coefficient.
5.7 Pyrometer (mh)
A pyrometer is a non-contacting device that intercepts and measures thermal radiation, a process known as primary. This device can be used to determine the temperature of an object's surface.
A temperature-measuring device, originally an instrument that measures temperatures beyond the range of thermometers, but now in addition a device that measures thermal radiation in any temperature range.
The illustration shows a very simple type of radiation pyrometer. Part of the thermal radiation emitted by a hot object is intercepted by a lens and focused onto a thermopile. The resultant heating of the thermopile causes it to generate an electrical signal (proportional to the thermal radiation) which can be displayed on a recorder. The figure below is of Elementary radiation pyrometer.
Unfortunately, the thermal radiation emitted by the object depends not only on its temperature
but also on its surface characteristics. The radiation existing inside hot, opaque objects is so-called blackbody radiation, which is a unique function of temperature and wavelength and is the same for all opaque materials. However, such radiation, when it attempts to escape from the object, is partly reflected at the surface. In order to use the output of the pyrometer as a measure of target temperature, the effect of the surface characteristics must be eliminated. A cavity can be formed in an opaque material and the pyrometer sighted on a small opening extending from the cavity to the surface. The opening has no surface reflection, since the surface has been eliminated. Such a source is called a blackbody source, and is said to have an emittance of 1.00. By attaching thermocouples to the black-body source, a curve of pyrometer output voltage versus blackbody temperature can be constructed.
Pyrometers can be classified generally into types requiring that the field of view be filled, such as narrow-band and total-radiation pyrometers; and types not requiring that the field of view be filled, such as optical and ratio pyrometers. The latter depend upon making some sort of comparison between two or more signals.
The optical pyrometer should more strictly be called the disappearing-filament pyrometer. In operation, an image of the target is focused in the plane of a wire that can be heated electrically. A rheostat is used to adjust the current through the wire until the wire blends into the image of the target (equal brightness condition), and the temperature is then read from a calibrated dial on the rheostat.
The ratio, or “two-color,” pyrometer makes measurements in two wavelength regions and electronically takes the ratio of these measurements. If the emittance is the same for both wavelengths, the emittance cancels out of the result, and the true temperature of the target is obtained. This so-called gray-body assumption is sufficiently valid in some cases so that the “color temperature” measured by a ratio pyrometer is close to the true temperature. 5.7.1 Principle of operation (h)
A pyrometer has an optical system and a detector. The optical system focuses the thermal radiation onto the detector. The output signal of the detector (temperature T) is related to the thermal radiation or irradiance j* of the target object through the Stefan–Boltzmann law, the constant of proportionality σ, called the Stefan-Boltzmann constant and the emissivity ε of the object.
J* = εσT4
This output is used to infer the object's temperature. Thus, there is no need for direct contact between the pyrometer and the object, as there is with thermocouples and resistance temperature detectors (RTDs).
5.7.2 Types of Pyrometer
A pyrometer measures heat admitted from an object visibly bright or incandescent. Pyrometers are a class of thermometers scientists use to determine the heat and type of heat emitted from an object. The crucial difference between a pyrometer and other types of thermometers is the incandescent levels from the heated objects are usually far too hot for contact. That is why pyrometers have optical scanners that measure the heat. Since there are different types and levels of heat, there are different types of pyrometers.
Broadband Pyrometer
A broadband pyrometer is one of the most-used pyrometers by scientists. The broadband pyrometer registers the broadband wavelengths of radiation, usually around 0.3 microns. Though most often used, they can have large errors in readings. Since they are only registering a small amount of heat from an object, everything from water vapor to dust can create a reading error.
Optical Pyrometers
Although all pyrometers are optical in the sense they can read an object's heat from a distance, an optical pyrometer allow a scientist to see heat. An optical pyrometer measures the infrared wavelengths of heat and directly shows the user the heat distribution of an object. Other pyrometers usually have a screen that provides the results of an optical scan.
An optical pyrometer is like a telescope wherein scientists can look through a lens and see the infrared wavelengths of an object. Optical pyrometers are one of the oldest pyrometer types and are able to see the wavelength levels up to 0.65 microns.
Radiation Pyrometer
A radiation pyrometer measures pure radiation wavelengths. The device has an optical scanner that can see 0.7 to 20 microns on the wavelength range, the general range for radioactive heat. The optical scanner helps scientists measure radiation levels without putting the pyrometer up to the object, which could endanger the individual to radiation exposure.
5.7.3 Applications (h)
Pyrometers are suited especially to the measurement of moving objects or any surfaces that can not be reached or can not be touched.
Smelter Industry-
Temperature is a fundamental parameter in metallurgical furnace operations. Reliable and continuous measurement of the melt temperature is essential for effective control of the operation. Smelting rates can be maximized, slag can be produced at the optimum temperature,
fuel consumption is minimized and refractory life may also be lengthened. Thermocouples were the traditional devices used for this purpose, but they are unsuitable for continuous measurement because they rapidly dissolve
Over-the-bath Pyrometer-
Salt bath furnaces operate at temperatures up to 1300 °C and are used for heat treatment. At very high working temperatures with intense heat transfer between the molten salt and the steel being treated, precision is maintained by measuring the temperature of the molten salt. Most errors are caused by slag on the surface which is cooler than the salt bath.
Tuyère Pyrometer-
The Tuyère Pyrometer is an optical instrument for temperature measurement through the tuyeres which are normally used for feeding air or reactants into the bath of the furnace.
Steam Boilers
A steam boiler may be fitted with a pyrometer to measure the steam temperature in the superheater.
Hot Air Ballons
A hot air balloon is equipped with a pyrometer for measuring the temperature at the top of the envelope in order to prevent overheating of the fabric.
5.8 Thermometer (mh)
A thermometer (from the Greek θερμός, thermos, meaning "hot" and μ�τρον, metron, "measure") is a device that measures temperature or temperature gradient using a variety of different principles. A thermometer has two important elements: the temperature sensor (e.g. the bulb on a mercury-in-glass thermometer) in which some physical change occurs with temperature, plus some means of converting this physical change into a numerical value (e.g. the visible scale that is marked on a mercury-in-glass thermometer).
5.8.1 Physical principles of thermometry (h)
Thermometers may be described as empirical or absolute. Absolute thermometers are calibrated numerically by the thermodynamic absolute temperature scale. Empirical thermometers are not in general necessarily in exact agreement with absolute thermometers as to their numerical scale readings, but to qualify as thermometers at all they must agree with absolute thermometers and with each other in the following way: given any two bodies isolated in their separate respective
thermodynamic equilibrium states, all thermometers agree as to which of the two has the higher temperature, or that the two have equal temperatures. For any two empirical thermometers, this does not require that the relation between their numerical scale readings be linear, but it does require that relation to be strictly monotonic. This is a fundamental character of temperature and thermometers.
There are several principles on which empirical thermometers are built, known as "Primary and secondary thermometers". Several such principles are essentially based on the constitutive relation between the state of a suitably selected particular material and its temperature. Only some materials are suitable for this purpose, and they may be considered as "thermometric materials". Radiometric thermometry, in contrast, can be only very slightly dependent on the constitutive relations of materials. In a sense then, radiometric thermometry might be thought of as "universal". This is because it rests mainly on a universality character of thermodynamic equilibrium, that it has the universal property of producing blackbodyradiation.
5.8.2 Thermometric Materials (h)
There are various kinds of empirical thermometer based on material properties.
Many empirical thermometers rely on the constitutive relation between pressure, volume and temperature of their thermometric material. For example, mercury expands when heated.
If it is used for its relation between pressure and volume and temperature, a thermometric material must have three properties:
1. Its heating and cooling must be rapid. That is to say, when a quantity of heat enters or leaves a body of the material, the material must expand or contract to its final volume or reach its final pressure and must reach its final temperature with practically no delay; some of the heat that enters can be considered to change the volume of the body at constant temperature, and is called the latent heat of expansion at constant temperature; and the rest of it can be considered to change the temperature of the body at constant volume, and is called the specific heat at constant volume. Some materials do not have this property, and take some time to distribute the heat between temperature and volume change.
2. Its heating and cooling must be reversible. That is to say, the material must be able to be heated and cooled indefinitely often by the same increment and decrement of heat, and still return to its original pressure, volume and temperature every time. Some plastics do not have this property.
3. Its heating and cooling must be monotonic. That is to say, throughout the range of temperatures for which it is intended to work,
(a) at a given fixed pressure, either
(α) the volume increases when the temperature increases, or else (β) the volume decreases when the temperature increases; not
(α) for some temperatures and (β) for others; or
(b) at a given fixed volume, either
(α) the pressure increases when the temperature increases, or else (β) the pressure decreases when the temperature increases; not
(α) for some temperatures and (β) for others.
At temperatures around about 4 °C, water does not have the property (3), and is said to behave anomalously in this respect; thus water cannot be used as a material for this kind of thermometry for temperature ranges near 4 °C.
Gases, on the other hand, all have the properties (1), (2), and (3)(a)(α) and (3)(b)(α). Consequently, they are suitable thermometric materials, and that is why they were important in the development of thermometry.
5.8.3 Primary & Secondary Thermometers (h)
Thermometers can be divided into two separate groups according to the level of knowledge about the physical basis of the underlying thermodynamic laws and quantities. For primary thermometers the measured property of matter is known so well that temperature can be calculated without any unknown quantities. Examples of these are thermometers based on the equation of state of a gas, on the velocity of sound in a gas, on the thermal noise, voltage or current of an electrical resistor, on blackbody radiation, and on the angular anisotropy of gamma ray emission of certain radioactive nuclei in amagnetic field. Primary thermometers are relatively complex.
Secondary thermometers are most widely used because of their convenience. Also, they are often much more sensitive than primary ones. For secondary thermometers knowledge of the measured property is not sufficient to allow direct calculation of temperature. They have to be calibrated against a primary thermometer at least at one temperature or at a number of fixed temperatures. Such fixed points, for example, triple points and superconducting transitions, occur reproducibly at the same temperature.
5.8.4 Uses (h)
Thermometers utilize a range of physical effects to measure temperature. Temperature sensors are used in a wide variety of scientific and engineering applications, especially measurement systems. Temperature systems are primarily either electrical or mechanical, occasionally inseparable from the system which they control (as in the case of a mercury-in-glass thermometer). Thermometers are used in roadways in cold weather climates to help determine if icing conditions exist. Indoors, thermistors are used in climate control systems such as air
conditioners, freezers, heaters, refrigerators, and water heaters. Galileo thermometers are used to measure indoor air temperature, due to their limited measurement range.
Alcohol thermometers, infrared thermometers, mercury-in-glass thermometers, recording thermometers, thermistors, and Six's thermometers are used in meteorology and climatology in various levels of the atmosphere and oceans. Aircraft use thermometers and hygrometers to determine if atmospheric icing conditions exist along their flight path. These measurements are used to initialize weather forecast models. Thermometers are used in roadways in cold weather climates to help determine if icing conditions exist and indoors in climate control systems.
Bi-metallic stemmed thermometers, thermocouples, infrared thermometers, and thermistors are handy during cooking in order to know if meat has been properly cooked. Temperature of food is important because if it sits in environments with a temperature between 5 and 57 °C (41 and 135 °F) for four hours or more, bacteria can multiply leading to foodborne illnesses. Thermometers are used in the production of candy.
Medical thermometers such as mercury-in-glass thermometers, infrared thermometers, pill thermometers, and liquid crystal thermometers are used in health care settings to determine if individuals have a fever or are hypothermic.
Thermochromic liquid crystals are also used in mood rings and in thermometers used to measure the temperature of water in fish tanks.
Fiber Bragg grating temperature sensors are used in nuclear power facilities to monitor reactor core temperatures and avoid the possibility of nuclear meltdowns.
A thermometer constructed for probing stored food is also called a "temperature wand".
5.9 Bi-Metallic Strip (mh)
A bimetallic strip is used to convert a temperature change into mechanical displacement. The strip consists of two strips of different metals which expand at different rates as they are heated, usually steel and copper, or in some cases steel and brass. The strips are joined together throughout their length by riveting, brazing or welding. The different expansions force the flat strip to bend one way if heated, and in the opposite direction if cooled below its initial temperature. The metal with the higher coefficient of thermal expansion is on the outer side of the curve when the strip is heated and on the inner side when cooled.
The sideways displacement of the strip is much larger than the small lengthways expansion in either of the two metals. This effect is used in a range of mechanical and electrical devices. In some applications the bimetal strip is used in the flat form. In others, it is wrapped into a coil for compactness. The greater length of the coiled version gives improved sensitivity.
5.9.1 Applications (h)
Clocks-
Mechanical clock mechanisms are sensitive to temperature changes which lead to errors in time keeping. A bimetallic strip is used to compensate for this in some mechanisms. The most common method is to use a bimetallic construction for the circular rim of the balance wheel. As the spring controlling the balance becomes weaker with increasing temperature, so the balance becomes smaller in diameter to keep the period of oscillation (and hence timekeeping) constant.
Thermometers-
A direct indicating dial thermometer (such as a patio thermometer or a meat thermometer) uses a bimetallic strip wrapped into a coil. One end of the coil is fixed to the housing of the device and the other drives an indicating needle. A bimetallic strip is also used in a recording thermometer.
Electrical devices-
Bimetal strips are used in miniature circuit breakers to protect circuits from excess current. A coil of wire is used to heat a bimetal strip, which bends and operates a linkage that unlatches a spring-operated contact. This interrupts the circuit and can be reset when the bimetal strip has cooled down.Bimetal strips are also used in time-delay relays, lamp flashers, and fluorescent lamp starters. In some devices the current running directly through the bimetal strip is sufficient to heat it and operate contacts directly.
Thermostats-
In the regulation of heating and cooling, thermostats that operate over a wide range of temperatures are used. In these, one end of the bimetal strip is mechanically fixed and attached to an electrical power source, while the other (moving) end carries an electrical contact. In adjustable thermostats another contact is positioned with a regulating knob or lever. The position so set controls the regulated temperature, called the set point.
Some thermostats use a mercury switch connected to both electrical leads. The angle of the entire mechanism is adjustable to control the set point of the thermostat.
Depending upon the application, a higher temperature may open a contact (as in a heater control) or it may close a contact (as in a refrigerator or air conditioner).
The electrical contacts may control the power directly (as in a household iron) or indirectly, switching electrical power through a relay or the supply ofnatural gas or fuel oil through an electrically operated valve. In some natural gas heaters the power may be provided with a thermocouple that is heated by a pilot light (a small, continuously burning, flame). In devices without pilot lights for ignition (as in most modern gas clothes dryers and some natural gas
heaters and decorative fireplaces) the power for the contacts is provided by reduced household electrical power that operates a relay controlling an electronic ignitor, either a resistance heater or an electrically powered spark generating device.
Heat Engines
Simple toys have been built which demonstrate how the principle can be used to drive a heat engine
5.10 Thermal Resistance (mh)
Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow (heat per time unit or thermal resistance). Thermal resistance is the reciprocal of thermal conductance.
• Thermal resistance R has the units (m2K)/W. • Specific thermal resistance or specific thermal resistivity Rλ in (K·m)/W is a material
constant. • Absolute thermal resistance Rth in K/W is a specific property of a component. It is e.g.,
a characteristic of a heat sink.
Absolute thermal resistance-
Absolute thermal resistance is the temperature difference across a structure when a unit of heat energy flows through it in unit time. It is the reciprocal of thermal conductance. The SI units of thermal resistance are kelvins per watt or the equivalent degrees Celsius per watt (the two are the same since as intervals Δ1 K = Δ1 °C).
The thermal resistance of materials is of great interest to electronic engineers because most electrical components generate heat and need to be cooled. Electronic components malfunction or fail if they overheat, and some parts routinely need measures taken in the design stage to prevent this.
5.11 Criteria for Selecting a Thermometer (mh)
1. The precision or resolution of a thermometer is simply to what fraction of a degree it is possible to make a reading. For high temperature work it may only be possible to measure to the nearest 10 °C or more. Clinical thermometers and many electronic thermometers are usually readable to 0.1 °C. Special instruments can give readings to one
thousandth of a degree. However, this precision does not mean the reading is true or accurate.
2. Thermometers which are calibrated to known fixed points (e.g. 0 and 100 °C) will be accurate (i.e. will give a true reading) at those points. Most thermometers are originally calibrated to a constant-volume gas thermometer.[citation needed] In between a process of interpolation is used, generally a linear one.[34] This may give significant differences between different types of thermometer at points far away from the fixed points. For example the expansion of mercury in a glass thermometer is slightly different from the change in resistance of a platinum resistance thermometer, so these two will disagree slightly at around 50 °C.[35] There may be other causes due to imperfections in the instrument, e.g. in a liquid-in-glass thermometer if the capillary tube varies in diameter.[
Example- How much heat (in calories) is needed to raise 20 grams of water from 5 oC to 40 oC?
Sol. Q = mc(Tf—Ti)
Q = x
m = 20 grams
c = 1 cal/goC (this is the specific heat of water)
Tf = 40 oC
Ti = 5
x = (20 g) (1 cal/goC)(40 -5 0C) = 700 cal
Example- A 5.00-kg ball of clay falls from a height of 6.50 meters. If 92.0% of its energy is converted into distortional work that flattens the clay upon impact, how much heat is produced when the clay strikes the ground?
Sol.The total energy at the top of the fall is converted into kinetic energy just before impact with the ground (h = 0), and the kinetic energy is transformed into work and heat when the ball actually strikes the ground. We could, therefore, calculate the kinetic energy from the velocity of the clay just before impact. (This velocity is 11 m/s if you decide to do this calculation for practice.) Because the total energy remains the same, however, it is easier to use conservation of energy and convert directly from potential energy into heat and work. The change in potential
energy of the clay must be equal to the work done to distort the clay plus any heat energy that the clay retains after impact.
The change in potential energy of the clay is given by:
Ep = m g h = 5.00 kg (9.80 m/s2)(6.50 m) = 319 J
This means a total of 319 J of energy is available from the fall. The work done to distort the clay on impact is given as 92.0%, but we must use the decimal equivalent 0.920 in our calculation. The work causing distortion is then negative because it is work done on the clay.
W = -0.920E = - 0.920 (319 J) = -293 J
The heat retained by the clay ball can then be found using the second law of thermodynamics:
H = E + W = 319 J + (-293 J) = 26 J
Thus we see that most of the energy goes to distort the molecular structure of the clay and only a small fraction of the energy causes an increase in the total internal KE of the molecules. The joule is a perfectly acceptable unit for heat, but we can also express the heat in kilocalories.
26 J (1 kcal / 4186 J) = 0.0062 kcal.
5.12 Review Questions (mh)
1. A child has a viral infection and his body temperature is 101.5° F. Find the corresponding temperature in °C.
2. Ethyl alcohol has a latent heat of vaporization of 200 cal/g. Find out what quantity would be necessary to condense to create enough heat to bring to boil 200 g of water from 98°C.
3. A silver teaspoon is placed in a hot cup of tea. The room temperature is 24.0°C and the tea is at 82.0°C. Silver has a coefficient of thermal expansion of 1.90 ·10–5°C– 1. In the process of thermal expansion, the change in length is measured to be 165 μm. Find the initial and final lengths of the teaspoon.
4. Consider two aluminum rods: Rod 1 has a length three times smaller than Rod 2 but the same size diameter. How do the rods change their length if they are subjected to the same change in temperature? Show your calculations.
5. An iron rod receives an amount of heat of 2,380 J from a heat reservoir. The rod has a mass of 0.500 kg. Find the temperature change.
CHAPTER-06
THERMAL EXPANSION OF SOLIDS (ch)
6.1 Thermal Expansion (mh)
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are unusual; this effect is limited in size, and only occurs within limited temperature ranges (see examples below). The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies with temperature.
6.1.1 Predicting Expansion (h)
If an equation of state is available, it can be used to predict the values of the thermal expansion at all the required temperatures and pressures, along with many other state functions.
6.1.2 Contraction Effects (h)
A number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than "thermal contraction". For example, the coefficient of thermal expansion of water drops to zero as it is cooled to 3.983 °C and then becomes negative below this temperature; this means that water has a maximum density at this temperature, and this leads to bodies of water maintaining this temperature at their lower depths during extended periods of sub-zero weather. Also, fairly pure silicon has a negative coefficient of thermal expansion for temperatures between about 18 and 120 Kelvin.
6.1.3 Factors Effecting Thermal Expansion (h)
Unlike gases or liquids, solid materials tend to keep their shape when undergoing thermal expansion.
Thermal expansion generally decreases with increasing bond energy, which also has an effect on the melting point of solids, so, high melting point materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids. The thermal expansion of glasses is higher compared to that of crystals. At the glass transition temperature, rearrangements that occur in an amorphous material lead to characteristic discontinuities of coefficient of thermal expansion or specific heat. These discontinuities allow detection of the glass transition temperature where a supercooled liquid transforms to a glass.
Absorption or desorption of water (or other solvents) can change the size of many common materials; many organic materials change size much more due to this effect than they do to thermal expansion. Common plastics exposed to water can, in the long term, expand by many percent.
6.2 Coefficient of Thermal Expansion (mh)
The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in size per degree change in temperature at a constant pressure. Several types of coefficients have been developed: volumetric, area, and linear. Which is used depending on the particular application and which dimensions are considered important. For solids, one might only be concerned with the change along a length, or over some area.
The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. In general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Substances that expand at the same rate in every direction are called isotropic. For isotropic materials, the area and linear coefficients may be calculated from the volumetric coefficient.
General volumetric thermal expansion coefficient-In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal expansion are given by
αv = 1/V(∆V/T)p
The subscript p indicates that the pressure is held constant during the expansion, and the subscript "V" stresses that it is the volumetric (not linear) expansion that enters this general definition. In the case of a gas, the fact that the pressure is held constant is important, because the volume of a gas will vary appreciably with pressure as well as temperature. For a gas of low density this can be seen from the ideal gas law.
6.3 Expansion in Solids (mh)
Materials generally change their size when subjected to a temperature change while the pressure is held constant. In the special case of solid materials, the pressure does not appreciably affect the size of an object, and so, for solids, it's usually not necessary to specify that the pressure be held constant.Common engineering solids usually have coefficients of thermal expansion that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, practical calculations can be based on a constant, average, value of the coefficient of expansion.
6.3.1 Linear expansion (h)
To a first approximation, the change in length measurements of an object ("linear dimension" as opposed to, e.g., volumetric dimension) due to thermal expansion is related to temperature change by a "linear expansion coefficient". It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write:
where L is a particular length measurement and dL/dT is the rate of change of that linear dimension per unit change in temperature.
The change in the linear dimension can be estimated to be:
This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature \Delta T. If it does, the equation must be integrated.
6.3.2 Effects on strain (h)
For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by \epsilon_\mathrm{thermal} and defined as:
where Linitial is the length before the change of temperature and Lfinal is the length after the change of temperature.For most solids, thermal expansion is proportional to the change in temperature:
Thus, the change in either the strain or temperature can be estimated by:
Where ∆T = (T final - Tinitial) is the difference of the temperature between the two recorded strains, measured in degrees Celsius or Kelvin, and αL is the linear coefficient of thermal expansion in "per degree Celcius" or "per Kelvin", denoted by °C−1 or K−1, respectively.
6.3.3 Area expansion (h)
The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write:
αA =(1/A)(dA/dT)
where A is some area of interest on the object, and dA/dT is the rate of change of that area per unit change in temperature.The change in the linear dimension can be estimated as:
∆A/A= αA ∆T
This equation works well as long as the linear expansion coefficient does not change much over the change in temperature ∆T. If it does, the equation must be integrated.
6.3.4 Volumetric expansion (h)
For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written:
Vα = 1/V(dV/Dt)
where V is the volume of the material, and dV/dT is the rate of change of that volume with temperature.This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2 %. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2 %. The volumetric expansion coefficient would be 0.2 % for 50 K, or 0.004 %/K.
If we already know the expansion coefficient, then we can calculate the change in volume
dV/V = VdT
where dV/V is the fractional change in volume (e.g., 0.002) and dT is the change in temperature (50° C).The above example assumes that the expansion coefficient did not change as the temperature changed. This is not always true, but for small changes in temperature, it is a good approximation.
Example- A steel rod has a length of exactly 20 cm at 30◦C. How much longer is it at50◦C? [Use αSteel = 11 ×10−6◦/C.]
Sol.The change in temperature of the steel rod is
∆T = 50◦
C −20◦
C = 30 C◦
and the length is 20.0 cm. Using the given value for the linear expansion coefficient α, wefind the change in length,
∆L = Lα ∆T = (20.0 cm)(11×10−6/◦C)(20.0C◦)=4.4×10−3cm.
The length of the bar increases by 4.4 ×10−3cm.
Example- An aluminum flagpole is 33 m high. By how much does its length increase as the temperature increases by 15 C◦?
[Use αAl = 23 ×10−6/C◦.]
Sol. the change in length of the (aluminum) flagpole is
∆L = Lα∆T = (33 m)(23 ×10−6/◦C)(15 C◦)=1.1 ×10−2m=1.1cm
Example- A certain substance has a mass per mole of 50 g/mole. When 314 J of heat is added to a 30.0 g sample of this material, its temperature rises from 25.0◦C to45.0◦C. (a) What is the specific heat of this substance? (b) How many moles of the substance are present? (c) What is the molar specific heat of the substance?
Sol.(a) From the relation between heat transferred and temperature change, Q = cm∆T, we solve for c and use the given data. (The mass of the sample is m = 0.0300 kg and the change in temperature is +20.0 K.)
c =Qm∆T
=(314 J)(0.0300 kg)(20.0 K) = 532 Jkg·K
The specific heat of the substance is 532 Jkg·K
(b) We are given molar mass of the substance, which is the correspondence between mass and moles:
50 g ←→ 1 mol
which we can use as a conversion factor to get:
30.0 g = (30.0 g)
(1.000 mol)(50.0 g)= 0.600 mol
So 0.600 mol of the sample is present.
(c) To get c as a molar specific heat, redo part (a) using moles instead of mass:
c =Qn∆T=(314 J)(0.600 kg)(20.0 K) = 26.6Jmol·K
The molar specific heat of the substance is 26.6calmol·K
6.4 Stress and Strain (mh)
In a deformed body, restoring force is set up within the body which tends to bring the body back to the normal position. The magnitude of these restoring force depends upon the deformationcaused. This restoring force per unit area of a deformed body is known as stress.
Stress = restoring force/areaN m–2
Its dimensional formula is ML–1T–2
Due to the application of deforming force, length, volume or shape of a body changes. Or in other words, the body is said to be strained. Thus, strain produced in a body is defined as the ratio of
change in dimension of a body to the original dimension.
Strain = change in dimension/original dimension
Strain is the ratio of two similar quantities. Therefore it has no unit.
6.4.1 Stress (h)
When some external forces are applied to a body, then the body offers internal resistance to these forces. This internal internal opposing force per unit area is called 'stress'. It is denoted by symbol σ and its S.I. unit is Pascal or N/m2. Mathematically the stress formula we can obtain as Stress = Force/Area 6.4.1.1 Types of Stress (sh)
Different types of stress : Stress is of two different types mainly (i) Normal Stress (ii) Shearing or Tangential Stress .
• Normal Stress : If the stress is normal to the surface, it is called normal stress. Stress is always normal in the case of a change in length or a wire or in the case of change in volume of a body
• Longitudinal Stress : When a normal stress change the length of a body then it is called longitudinal stress which is given by
Longitudinal Stress = Deforming Force/Area of cross section = F/A The longitudinal stress can be further divided into two types. When a wire or a rod is stretched at the two ends by equal and opposite forces, the stress is called tensile stress. When a rod is pushed at the two ends by equal and opposite forces, it will be under compression. The stress in such a case is called compressive stress. The pillars of a building experience compressive stress.
• Volume Stress (or ) Bulk Stress : When a normal stress changes the volume of a body then it is called volume stress. When a solid body is immersed in a fluid, the force at any point is normal to the surface of the body and the magnitude of the force on any small area is proportional to the area i.e., the body is under the action of a pressure P.
Bulk Stress = Force/Area = Pressure
• Shearing Stress : When the Stress is tangential to the surface due to the application of forces parallel to the surface, then the stress is called tangential or shearing stress. It changes the shape of the body.
Shearing Stress = Force/Surface Area = F/A
6.4.1.2 Tensile Stress (sh)
We all know what is meant by the term stress, Stress is a mechanical term which is defined as the average amount of total force applied on an unit area. In other words, it is a type of internal resistance of any type of material offered in order to get deformed and is measured in terms of load applied on the body. There are three categories of stress:
1. Compressive stress - This particular category of stress results in the compaction of the material.
2. Tensile stress - This type of stress results in the expansion of the material 3. Yield stress - The type of stress due to which their is plastic deformation of the body is
known as yield stress.
So, now we are quite familiar with the stress. Let us know something more about this topic. The another question which come to our mind is what causes stress? Stress is mainly produced due to the movement of internal particles of the material whenever we apply external forces on that particular material. Due to the internal motion of the particles internal forces are generated which is distributed uniformly throughout the body.
The tensile stress is represented by the symbol σ = F/A where, σ = Tensile stress ( N / mm2 ) F = Applied external force ( N ) A = cross sectional area ( mm2 ) The Units of tensile Stress comes out to be N / mm2 .
6.4.1.3 Elastic limit (sh)
If an elastic material is stretched or compressed beyond a certain limit, it will not regain its original state and will remain deformed. The limit beyond which permanent deformation occurs is called the elastic limit.
6.4.2 Strain (h)
Normal stress on a body causes change in length or volume and tangential stress produces change in shape of the body. The ratio of change produced in the dimensions of a body by a system of forces or couples, in equilibrium, to its original dimensions is called strain.
Strain = Change in Dimension/Original Dimension
6.4.2.1 Types of Strain (sh)
Strain is of three types depending upon the change produced in a body and the stress applied. The three types of strain are
1. Longitudinal Strain
2. Volume Strain and 3. Shearing Strain
• Longitudinal Strain : It is the ratio of the change in length of a body to the original length of the body. If L is the original length of a wire or a rod and the final length of the wire or the rod is L+ΔL under the action of a normal stress, the change in length is ΔL.
Longitudinal Strain = Change in length/Original length = ΔL/L
If the length increases due to tensile stress, the corresponding strain is called tensile strain. If the length decreases due to compressive stress, the strain is called compressive strain.
• Volume Strain : It is the ratio of the change in volume of a body to its original volume. If V is the original volume of a body and V +ΔV is the volume of the body under the action of a normal stress, the change in volume is ΔV.
Volume Strain = Change in volumeOriginal volume = ΔVV.
• Shearing Strain : If is the angle through which a face originally perpendicular to the fixed face is turned. (or) It is the ratio of the displacement of a layer to its distance from the fixed layer.
6.4.2.2 Stress Strain Equation (sh)
Hooke's law gives a relationship between the stress and strain. According to Hooke's law, within the elastic limit, strain produced in a body is directly proportional to the stress produced.
stress α strain
Stress/Strain = a constant, known as Modulus of Elasticity. Its unit is Nm-2.
6.5 Stress and Stress Relationship (mh)
During testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the sample, and strain, derived from measuring the deformation of the sample, i.e. elongation, compression, or distortion. The nature of the curve varies from material to material.
If we begin from origin and follows the graph a number of points are indicated.
1. Point A: At origin, there is no initial stress or strain in the test peice. Upto point A Hooke's Law is obeyed according to which Stress is directly proportional to Strain. That's why the point A is also known as proportional limit. This straight line region is known as Elastic Region and the material can regain its original shape after removal of load.
2. Point B: The portion of the curve between AB is not a straight line and Strain increases faster than stress at all points on the curve beyond point A. Point B is the point after which any continuous stress results in permanent, or inelastic deformation. Thus, point B is known as the elastic limit or yield point.
3. Point C & D: Beyond the point B, the material goes to the plastic stage till the point C is reached. At this point the cross- sectional area of the material starts decreasing and the stress decreases to point D. At point D the workforce changes its length with a little or without any increase in stress upto point E.
4. Point E: Point E indicates the location of the value of the ultimate stress. The portion DE is called the yielding of the material at constant stress. From point E on-wards, the strength of the material increases and requires more stress for deformation, until point F is reached.
5. Point F: A material is considered to have completely failed once it reaches the ultimate stress. The point of fracture, or the actual tearing of the material, does not occur until point F. The point F is also called Ultimate Point or Fracture Point.
Features of Stress Strain Curve Features of Stress Strain Curve vary for different types of materials i.e. Ductile and Brittle.
1. Features for Ductile Material: The capacity of being drawn out plastically (permanently) before fracture is called the ductility of the material. In case of ductility the material obeys the Hooke’s Law and takes time for fracture. In ductile material stress strain curve the plastic region is long and material will bear more strain before Fracture.
2. Features for Brittle Material: Materials which show very small or negligible elongation before they fracture are called brittle materials for e.g. Cast Iron, Tool steel, concrete etc. They get fractured in two or more parts without any prior notice. In brittle material stress strain curve the plastic region is small and the strength is high.
6.6 Review Questions (mh)
1. A brass rod is to be used in an application requiring its ends to be held rigid. If the rod is stress free at room temperature [20⁰C (68F)], what is the maximum temperature to which the rod may be heated without exceeding a compressive stress of 172 MPa (25,000 psi)? Assume a modulus of elasticity of 100 GPa (14.6 _ 106 psi) for brass.
2. To what temperature would 10 lbm of a brass specimen at 25⁰C (77F) be raised if 65 Btu of heat is supplied?
3. Estimate the energy required to raise the temperature of 2 kg (4.42 lbm) of the following materials from 20 to 100⁰C (68 to 212F): aluminum, steel, soda–lime glass, and high density polyethylene.
4. (a) Briefly explain why porosity decreases the thermal conductivity of ceramic and polymeric materials, rendering them more thermally insulative. (b) Briefly explain how the degree of crystallinity affects the thermal conductivity of polymeric materials and why.
5. The two ends of a cylindrical rod of nickel 120.00 mm long and 12.000 mm in diameter are maintained rigid. If the rod is initially at 70⁰C, to what temperature must it be cooled in order to have a 0.023-mm reduction in diameter?
6. For each of the following pairs of materials, decide which has the larger thermal conductivity. Justify your choices.
(a) Pure silver; sterling silver (92.5 wt%Ag–7.5 wt% Cu).
(b) Fused silica; polycrystalline silica.
(c) Linear and syndiotactic poly(vinyl chloride)(nn=1000); linear and syndiotactic polystyrene (nn=1000).
(d) Atactic polypropylene (Mw=106 g/mol); isotactic polypropylene (Mw=105 g/mol).
7. For some ceramic materials, why does the thermal conductivity first decrease and then increase with rising temperature?
CHAPTER-07
HEAT TRANSFER (ch)
7.1 Heat Transfer (mh)
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy and heat between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system.
Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature, as described by the second law of thermodynamics.
Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. The latter process is often called "natural convection". All convective processes also move heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is forced to flow by use of a pump, fan or other mechanical means.
Heat is defined in physics as the transfer of thermal energy across a well-defined boundary around a thermodynamic system. It is a characteristic of a process and is never contained in matter. In engineering contexts, however, the term heat transfer has acquired a specific usage, despite its literal redundancy of the characterization of transfer. In these contexts, heat is taken as synonymous to thermal energy. This usage has its origin in the historical interpretation of heat as a fluid (caloric) that can be transferred by various causes, and that is also common in the language of laymen and everyday life.
7.1.1 Rate of Heat Transfer (h)
The rate at which heat is transferred or conducted through a substance is directly proportional to the
1. Area of the surface (A) perpendicular to the flow of heat and 2. The temperature gradient ΔT/x along the path of heat transfer.
For a one dimensional steady state heat transfer Rate of Heat Transfer is expressed by Fourier equation: Q = kA ΔT/Δx. where, K = Thermal Conductivity depends on the material A = Area of the surface ΔT/Δx = Temperature gradient for small change in temperature with respect to distance.
7.1.2 Types of Heat Transfer (h)
There are three modes of Heat transfer:
1. Conduction 2. Convection 3. Radiation
Conduction: Conduction is a process where the heat transfer takes place between the two solid bodies in contact, two regions of the same solid body. This will happen because of the hot, vibrating, and rapidly moving molecules transfer the heat to their neighboring atoms. Convection: The convection is a type of heat transfer where the heat transfer takes place through a medium and the medium may be liquid or the gas. The heat transfer takes place by the movement of fluid from one place to another. The heat transfer here is due to the bulk motion of the fluid.
Convection is described by the Newton’s law of cooling; the law states that the rate of heat loosed by a body is proportional to the difference in temperatures between the body and its surroundings.
Radiation: The Third mode of energy transfer is Radiation Heat Transfer. The transfer of heat from hot body to a cold body with any material medium for propagation. Every object in the universe is
made up of atoms and molecules. These atoms and molecules vibrate due to thermal energy present in them. Every object emits electromagnetic radiations because of the thermal vibrations of these atoms and molecules. In case of energy transfer, the radiation conversion of radiated electromagnetic energy to thermal energy takes place. Radiation Heat Transfer can also be termed as transfer of energy through waves.
7.1.3 Heat Transfer Coefficient (h)
It is defined as the ratio of heat lost due to the flow to the product of area and temperature difference. It is used to measure the transfer of heat by convection or in the phase change (generally between fluid and a solid).
h = Q/AΔT
Where,
Q = Heat flow in input or lost heat flow, J/s = W
h = Heat transfer coefficient, W/m2/K.
A = Heat transfer surface area, m2
Δ T = difference in temperature between the solid surface and surrounding fluid area,
Heat flux is the heat flowing per unit area – Q/A.
Convective Heat Transfer coefficient of Air - hair= 10 to 100 W/(m2K).
Heat Transfer coefficient of Water - hwater = 500 to 10,000 W/(m2K).
7.1.4 Heat Transfer Example (h)
Examples of Heat Transfer through Conduction:
Take one long piece of metal. Put first end of this metal in the flame. Gradually, you will find that the temperature of the other end of the metal in your hand starts increasing. Energy gets transferred from the first end under flame to the second end of the metal in your hand by conduction. Examples of Heat Transfer through Natural Convection:
Natural convection is the convection which occurs naturally due to the bouncy effect. When water is heated in a pot then the particles, atoms or molecules of the water which are in contact
with the pot gain energy. Kinetic energy of these particles gets increased. As a result, density of these particles decreases and they start moving upward towards the open surface of the port. Cool water which is in contact with the air in the port starts sinking downward and convection current gets established. Cool water in the upper surface is denser than the hot water in contact with the surface of the pot. Convection Currents flow in circular fashion. Therefore, heat is transferred by the movement of water (fluids).
Forced convection: The convection that takes place by the force that is the force created from the fans, stirrers, pumps and etc.
Examples of Heat Transfer through Radiation:
Radiations coming out from the burner of the electric stove or toaster coils. Transfer of heat energy from Sun to Earth or to us takes place due to the electromagnetic rays emitted by the sun. Another example is the heating of food inside the microwave.
7.2 Modes of Heat Transfer (mh)
The fundamental modes of heat transfer are: Conduction or diffusion. The transfer of energy between objects that are in physical contact.
7.2.1 Convection (h)
The transfer of energy between an object and its environment, due to fluid motion.
7.2.2 Radiation (h)
The transfer of energy to or from a body by means of the emission or absorption of electromagnetic radiation.
7.2.3 Advection (h)
The transfer of energy from one location to another as a side effect of physically moving an object containing that energy.
7.2.4 Conduction (h)
On a microscopic scale, heat conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring particles. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. Fluids—especially gases—are less conductive. Thermal contact conductance
is the study of heat conduction between solid bodies in contact.Steady state conduction (see Fourier's law) is a form of conduction that happens when the temperature difference driving the conduction is constant, so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further. In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out.Transient conduction occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories or numerical analysis by computer.
7.2.5 Convection (h)
Convective heat transfer, or convection, is the transfer of heat from one place to another by the movement of fluids, a process that is essentially the transfer of heat via mass transfer. Bulk motion of fluid enhances heat transfer in many physical situations, such as (for example) between a solid surface and the fluid. Convection is usually the dominant form of heat transfer in liquids and gases. Although sometimes discussed as a third method of heat transfer, convection is usually used to describe the combined effects of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming.The process of transport by fluid streaming is known as advection, but pure advection is a term that is generally associated only with mass transport in fluids, such as advection of pebbles in a river. In the case of heat transfer in fluids, where transport by advection in a fluid is always also accompanied by transport via heat diffusion (also known as heat conduction) the process of heat convection is understood to refer to the sum of heat transport by advection and diffusion/conduction.Free, or natural, convection occurs when bulk fluid motions (steams and currents) are caused by buoyancy forces that result from density variations due to variations of temperature in the fluid. Forced convection is a term used when the streams and currents in the fluid are induced by external means—such as fans, stirrers, and pumps—creating an artificially induced convection current.Convective heating or cooling in some circumstances may be described by Newton's law of cooling: "The rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings." However, by definition, the validity of Newton's law of cooling requires that the rate of heat loss from convection be a linear function of ("proportional to") the temperature difference that drives heat transfer, and in convective cooling this is sometimes not the case. In general, convection is not linearly dependent on temperature gradients, and in some cases is strongly nonlinear. In these cases, Newton's law does not apply.
7.2.6 Radiation (h)
Red-hot iron object, transferring heat to the surrounding environment primarily through thermal radiation.Thermal radiation is energy emitted by matter as electromagnetic waves, due to the pool of thermal energy in all matter with a temperature above absolute zero. Thermal radiation propagates without the presence of matter through the vacuum of space.Thermal radiation is a
direct result of the random movements of atoms and molecules in matter. Since these atoms and molecules are composed of charged particles (protons and electrons), their movement results in the emission of electromagnetic radiation, which carries energy away from the surface.Unlike conductive and convective forms of heat transfer, thermal radiation can be concentrated in a small spot by using reflecting mirrors, which is exploited in concentrating solar power generation. For example, the sunlight reflected from mirrors heats the PS10 solar power tower and during the day it can heat water to 285 °C (545 °F).
7.3 Modeling Approaches of Heat Transfer (mh)
Complex heat transfer phenomena can be modeled in different ways.
7.3.1 Heat Equation (h)
The heat equation is an important partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. In some cases, exact solutions of the equation are available; in other cases the equation must be solved numerically using computational methods. For example, simplified climate models may use Newtonian cooling, instead of a full (and computationally expensive) radiation code, to maintain atmospheric temperatures.
7.3.2 Lumped System Analysis (h)
System analysis by the lumped capacitance model is a common approximation in transient conduction that may be used whenever heat conduction within an object is much faster than heat conduction across the boundary of the object.
This is a method of approximation that reduces one aspect of the transient conduction system—that within the object—to an equivalent steady state system. That is, the method assumes that the temperature within the object is completely uniform, although its value may be changing in time.
In this method, the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the object's boundary, known as the Biot number, is calculated. For small Biot numbers, the approximation of spatially uniform temperature within the object can be used: it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.
Lumped system analysis often reduces the complexity of the equations to one first-order linear differential equation, in which case heating and cooling are described by a simple exponential solution, often referred to as Newton's law of cooling.
7.4 Applications & Techniques of Heat Transfer (mh)
Heat transfer has broad application to the functioning of numerous devices and systems. Heat-transfer principles may be used to preserve, increase, or decrease temperature in a wide variety of circumstances.
7.4.1 Insulation & Radiant Barriers (h)
Thermal insulators are materials specifically designed to reduce the flow of heat by limiting conduction, convection, or both. Radiant barriers are materials that reflect radiation, and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice versa. Metal, for instance, is an excellent reflector and a poor insulator.
The effectiveness of an insulator is indicated by its R-value, or resistance value. The R-value of a material is the inverse of the conduction coefficient (k) multiplied by the thickness (d) of the insulator. In most of the world, R-values are measured in SI units: square-meter kelvins per watt (m²·K/W). In the United States, R-values are customarily given in units of British thermal units per hour per square-foot degrees Fahrenheit (Btu/h·ft²·°F).
Rigid fiberglass, a common insulation material, has an R-value of four per inch, while poured concrete, a poor insulator, has an R-value of 0.08 per inch.
The tog is a measure of thermal resistance, commonly used in the textile industry, and often seen quoted on, for example, duvets and carpet underlay.
The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa. At any specific wavelength, reflectivity = 1 - emissivity. An ideal radiant barrier would have a reflectivity of 1, and would therefore reflect 100 percent of incoming radiation. Vacuum flasks, or Dewars, are silvered to approach this ideal. In the vacuum of space, satellites use multi-layer insulation, which consists of many layers of aluminized (shiny) Mylar to greatly reduce radiation heat transfer and control satellite temperature.
7.4.2 Critical Insulation Thickness (h)
Low thermal conductivity (k) materials reduce heat fluxes. The smaller the k value, the larger the corresponding thermal resistance (R) value. Thermal conductivity is measured in watts-per-meter
per kelvin (W·m−1·K−1), represented as k. As the thickness of insulating material increases, the thermal resistance—or R-value—also increases.
However, adding layers of insulation to curved surfaces, such as cylinders and spheres, increases the surface area for convection. This may actually increase the heat transfer. This does not apply for flat surfaces.
Consider the following example. As insulation is added to the external surface of a thin cylinder, the outer radius of the pipe-and-insulation system increases, increasing the outer surface area. For small amounts of insulation the heat transfer may actually increase with added insulation. There is a critical radius (from the center of the cylinder to the outer edge of the insulation) at which the heat transfer is a maximum. Above this critical radius, added insulation decreases the heat transfer. For insulated cylinders, the critical radius is given by the equation
This equation shows that the critical radius depends only on the heat transfer coefficient and the thermal conductivity of the insulation, it does not depend on the radius of the cylinder. If the radius of the uninsulated cylinder is larger than the critical radius for insulation, the addition of any amount of insulation will decrease the heat transfer.
7.4.3 Heat Exchangers (h)
A heat exchanger is a tool built for efficient heat transfer from one fluid to another, whether the fluids are separated by a solid wall so that they never mix, or the fluids are in direct contact. Heat exchangers are widely used in refrigeration, air conditioning, space heating, power generation, and chemical processing. One common example of a heat exchanger is a car's radiator, in which the hot coolant fluid is cooled by the flow of air over the radiator's surface.
Common types of heat exchanger flows include parallel flow, counter flow, and cross flow. In parallel flow, both fluids move in the same direction while transferring heat; in counter flow, the fluids move in opposite directions; and in cross flow, the fluids move at right angles to each other. Common constructions for heat exchanger include shell and tube, double pipe, extruded finned pipe, spiral fin pipe, u-tube, and stacked plate.
When engineers calculate the theoretical heat transfer in a heat exchanger, they must contend with the fact that the driving temperature difference between the two fluids varies with position. To account for this in simple systems, the log mean temperature difference (LMTD) is often used as an "average" temperature. In more complex systems, direct knowledge of the LMTD is not available, and the number of transfer units (NTU) method can be used instead.
7.4.4 Heat Dissipation (h)
A heat sink is a component that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in
refrigeration and air conditioning systems, and the radiator in a car (which is also a heat exchanger). Heat sinks also help to cool electronic and optoelectronic devices such as CPUs, higher-power lasers, and light-emitting diodes (LEDs). A heat sink uses its extended surfaces to increase the surface area in contact with the cooling fluid.
7.4.5 Buildings (h)
In cold climates, houses with their heating systems form dissipative systems, often resulting in a loss of energy (known colloquially as "Heat Bleed") that makes home interiors uncomfortably cool or cold.
For the comfort of the inhabitants, the interiors must be maintained out of thermal equilibrium with the external surroundings. In effect, these domestic residences are islands of warmth in a sea of cold, and the thermal gradient between the inside and outside is often quite steep. This can lead to problems such as condensation and uncomfortable air currents, which—if left unaddressed—can cause cosmetic or structural damage to the property.
Such issues can be prevented through the execution of an energy audit, and the implementation of recommended corrective procedures (such as the installation of adequate insulation, the air sealing of structural leaks, and the addition of energy-efficient windows and doors.
Thermal transmittance is the rate of transfer of heat through a structure divided by the difference in temperature across the structure. It is expressed in watts per square meter per kelvin, or W/m²K. Well-insulated parts of a building have a low thermal transmittance, whereas poorly-insulated parts of a building have a high thermal transmittance.
A thermostat is a device capable of starting the heating system when the house's interior falls below a set temperature, and of stopping that same system when another (higher) set temperature has been achieved. Thus, the thermostat controls the flow of energy into the house, that energy eventually being dissipated to the exterior.
7.4.6 Heat Transfer in the Human Body (h)
The principles of heat transfer in engineering systems can be applied to the human body in order to determine how the body transfers heat. Heat is produced in the body by the continuous metabolism of nutrients which provides energy for the systems of the body. The human body must maintain a consistent internal temperature in order to maintain healthy bodily functions. Therefore, excess heat must be dissipated from the body to keep it from overheating. When a person engages in elevated levels of physical activity, the body requires additional fuel which increases the metabolic rate and the rate of heat production. The body must then use additional methods to remove the additional heat produced in order to keep the internal temperature at a healthy level.
Heat transfer by convection is driven by the movement of fluids over the surface of the body. This convective fluid can be either a liquid or a gas. For heat transfer from the outer surface of
the body, the convection mechanism is dependent on the surface area of the body, the velocity of the air, and the temperature gradient between the surface of the skin and the ambient air.[21] The normal temperature of the body is approximately 37°C. Heat transfer occurs more readily when the temperature of the surroundings is significantly less than the normal body temperature. This concept explains why a person feels “cold” when not enough covering is worn when exposed to a cold environment. Clothing can be considered an insulator which provides thermal resistance to heat flow over the covered portion of the body. This thermal resistance causes the temperature on the surface of the clothing to be less than the temperature on the surface of the skin. This smaller temperature gradient between the surface temperature and the ambient temperature will cause a lower rate of heat transfer than if the skin were not covered.
In order to ensure that one portion of the body is not significantly hotter than another portion, heat must be distributed evenly through the bodily tissues. Blood flowing through blood vessels acts as a convective fluid and helps to prevent any buildup of excess heat inside the tissues of the body. This flow of blood through the vessels can be modeled as pipe flow in an engineering system. The heat carried by the blood is determined by the temperature of the surrounding tissue, the diameter of the blood vessel, the thickness of the fluid, velocity of the flow, and the heat transfer coefficient of the blood. The velocity, blood vessel diameter, and the fluid thickness can all be related with the Reynolds Number, a dimensionless number used in fluid mechanics to characterize the flow of fluids.
7.4.7 Others (h)
A heat pipe is a passive device constructed in such a way that it acts as though it has extremely high thermal conductivity. Heat pipes use latent heat and capillary action to move heat, and can carry many times as much heat as a similar-sized copper rod. Originally invented for use in satellites, they have applications in personal computers. Another major use is for solar hot water panels.
A thermocouple is a junction between two different metals that produces a voltage related to a temperature difference. Thermocouples are a widely used type of temperature sensor for measurement and control, and can also be used to convert heat into electric power.
A thermopile is an electronic device that converts thermal energy into electrical energy. It is composed of thermocouples. Thermopiles do not measure the absolute temperature, but generate an output voltage proportional to a temperature difference. Thermopiles are widely used, e.g., they are the key component of infrared thermometers, such as those used to measure body temperature via the ear.
A thermoelectric cooler is an solid state electronic device that pumps (transfers) heat from one side of the device to the other when electrical current is passed through it. It is based on the Peltier effect.
A thermal diode or thermal rectifier is a device that preferentially passes heat in one direction: a "one-way valve" for heat.
7.5 Thermal Conductivity (mh)
In physics, thermal conductivity (often denoted k, λ, or κ) is the property of a material to conduct heat. It is evaluated primarily in terms of Fourier's Law for heat conduction.Heat transfer occurs at a higher rate across materials of high thermal conductivity than across materials of low thermal conductivity. Correspondingly materials of high thermal conductivity are widely used in heat sink applications and materials of low thermal conductivity are used as thermal insulation. Thermal conductivity of materials is temperature dependent. The reciprocal of thermal conductivity is called thermal resistivity.
7.5.1 Determine the coefficient of thermal conductivity of a bad conductor using Lee's disc apparatus (h)
Theory:
The Lee’s Disc experiment determines an approximate value for the thermal conductivity k of a poor conductor like glass, cardboard, etc. The procedure is to place a disc made of the poor conductor, radius r and thickness x, between a steam chamber and two good conductivity metal discs (of the same metal) and allow the setup to come to equilibrium, so that the heat lost by the lower disc to convection is the same as the heat flow through the poorly conducting disc. The upper disc temperature T2 and the lower disc temperature T1 are recorded. The poor conductor is removed and the lower metal disc is allowed to heat up to the upper disc temperature T2. Finally, the steam chamber and upper disc are removed and replaced by a disc made of a good insulator.
The metal disc is then allowed to cool through T1 < T2 and toward room temperature T0. The temperature of the metal disc is recorded as it cools so a cooling curve can be plotted. Then the slope s1 =ΔT/Δt of the cooling curve is measured graphically where the curve passes through temperature T1.
At the steady state, rate of heat transfer (H) by conduction is given by;
Where,
k - Thermal conductivity of the sample
A - Cross sectional area,
T2 - T1 -Temperature difference across the sample.
x -Thickness of the bad conductor (see figure 1)
The sample is an insulator . It is in the form of a thin disc with large cross sectional area (A = πr2) compared to the area exposed at the edge (a = 2πrx) in order to reduce the energy loss. Rate of energy transfer across the sample can be increased by keeping 'x 'small and 'A 'large. Keeping x small means the apparatus will reach a steady state quickly.
The thin sample of disc is sandwiched between the brass disc and brass base of the steam chamber (see figure 2). The temperature of the brass disc is measured by thermometer T1 and the temperature of the brass base is measured by thermometer T2 . In this way the temperature difference across such a thin disc of sample can be accurately measured.
The temperatures T1 and T2 are constant when the apparatus is in steady state. Then the rate of heat conducted through the brass disc must be equal to the rate of heat loss due to cooling from the bottom of the brass disc by air convection. By measuring how fast the brass disc cools at the steady state temperature T1, the rate of heat loss can be determined. It shown in figure 3. If the disc cools down at a rate, then the rate of heat loss is given by:
Where,
m- mass of the brass disc
c - specific heat capacity of brass.
At steady state, heat conducted through the bad conductor per second will be equal to heat radiated per second from the exposed portion of the metallic disc.
Where,
k - Coefficient of thermal conductivity of the sample,
A - Area of the sample in contact with the metallic disc,
x - Thickness of the sample,
T2- T1 -Temperature difference across the sample thickness,
m - Mass of the metallic disc,
c - The Heat capacity of the metallic disc,
dT/dt - Rate of cooling of the metallic disc at T2.
7.5.2 Determination of thermal conductivity by searle’s bar method (h)
Searle's bar method is an experimental procedure to measure thermal conductivity of material. A bar of material is being heated by steam on one side and the other side cooled down by water
while the length of the bar is thermally insulated. Then the heat ΔQ propagating through the bar in a time interval of Δt is given by
dQ / dT = -KAdT/T
Where, ΔQ is the heat supplied to the bar in time Δt
k is the coefficient of thermal conductivity of the bar.
A is the cross-sectional area of the bar,
ΔTbar is the temperature difference of both ends of the bar
L is the length of the bar
and the heat ΔQ absorbed by water in a time interval of Δt is:
dQ/dT= CW dm/dt*dTW
Where, Cw is the specific heat of water,
Δm is the mass of water collected during time Δt,
ΔT water is difference in the temperature of water before and after it has gone through the bar.
Assuming perfect insulation and no energy loss, then
(dQ/dT)bar= (dQ/dT)water
7.6 Prevost’s Theory of Exchange (mh)
Prevost suggested that all bodies radiate energy, but hotter bodies radiate more heat than colder bodies. Suppose for example that we have two bodies 1 and 2 at different temperatures and with Each body will then radiate heat to the other but since the energy radiated from 1 to 2 is greater than the energy radiated from 2 to 1.
Heat will be transferred from body 1 to body 2 as long as Eventually the bodies will be at the same temperature soand heat will be transferred from body 1 to 2 at the same rate as from 2 to 1. The bodies will then be in thermal equilibrium.
7.7 Heat Radiation (mh)
Heat radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The mechanism is that bodies with a temperature above absolute zero have atoms or molecules with kinetic energies which are changing, and these changes result in charge-acceleration and/or dipole oscillation of the charges that compose the atoms. This motion of charges produces electromagnetic radiation in the usual way. However, the wide spectrum of this radiation reflects the wide spectrum of energies and accelerations of the charges in any piece of matter at even a single temperature.
Examples of thermal radiation include the visible light and infrared light emitted by an incandescent light bulb, the infrared radiation emitted by animals and detectable with an infrared camera, and the cosmic microwave background radiation. Thermal radiation is different from thermal convection and thermal conduction—a person near a raging bonfire feels radiant heating from the fire, even if the surrounding air is very cold.
7.7.1 Characteristics of heat radiation (h)
There are four main properties that characterize thermal radiation (in the limit of the far field):
1. Thermal radiation emitted by a body at any temperature consists of a wide range of frequencies. The frequency distribution is given by Planck's law of black-body radiation for an idealized emitter. This is shown in the right-hand diagram.
2. The dominant frequency (or color) range of the emitted radiation shifts to higher frequencies as the temperature of the emitter increases. For example, a red hot object radiates mainly in the long wavelengths (red and orange) of the visible band. If it is heated further, it also begins to emit discernible amounts of green and blue light, and the
spread of frequencies in the entire visible range cause it to appear white to the human eye; it is white hot. However, even at a white-hot temperature of 2000 K, 99% of the energy of the radiation is still in the infrared. This is determined by Wien's displacement law. In the diagram the peak value for each curve moves to the left as the temperature increases.
3. The total amount of radiation of all frequencies increases steeply as the temperature rises; it grows as T4, where T is the absolute temperature of the body. An object at the temperature of a kitchen oven, about twice the room temperature on the absolute temperature scale (600 K vs. 300 K) radiates 16 times as much power per unit area. An object at the temperature of the filament in an incandescent light bulb—roughly 3000 K, or 10 times room temperature—radiates 10,000 times as much energy per unit area. The total radiative intensity of a black body rises as the fourth power of the absolute temperature, as expressed by the Stefan–Boltzmann law. In the plot, the area under each curve grows rapidly as the temperature increases.
4. The rate of electromagnetic radiation emitted at a given frequency is proportional to the amount of absorption that it would experience by the source. Thus, a surface that absorbs more red light thermally radiates more red light. This principle applies to all properties of the wave, including wavelength (color), direction, polarization, and even coherence, so that it is quite possible to have thermal radiation which is polarized, coherent, and directional, though polarized and coherent forms are fairly rare in nature far from sources (in terms of wavelength). See section below for more on this qualification.
7.7.2 Radiative Heat Transfer (h)
The radiative heat transfer from one surface to another is equal to the radiation entering the first surface from the other, minus the radiation leaving the first surface.
• For a black body
Using the reciprocity rule, , this simplifies to:
where is the Stefan–Boltzmann constant and is the view factor from surface 1 to surface 2.
• For a grey body with only two surfaces the heat transfer is equal to:
where are the respective emissivities of each surface. However, this value can easily change for different circumstances and different equations should be used on a case per case basis.
7.8 Blackbody Radiation (mh)
Black-body radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.
A perfectly insulated enclosure that is in thermal equilibrium internally contains black-body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have negligible effect upon the equilibrium.
A black-body at room temperature appears black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. At higher temperatures, black bodies glow with increasing intensity and colors that range from dull red to blindingly brilliant blue-white as the temperature increases.
Although planets and stars are neither in thermal equilibrium with their surroundings nor perfect black bodies, black-body radiation is used as a first approximation for the energy they emit. Black holes are near-perfect black bodies, and it is believed that they emit black-body radiation (called Hawking radiation), with a temperature that depends on the mass of the hole.
The term black body was introduced by Gustav Kirchhoff in 1860. When used as a compound adjective, the term is typically written as hyphenated, for example, black-body radiation, but sometimes also as one word, as in blackbody radiation. Black-body radiation is also called complete radiation or temperature radiation or thermal radiation.
7.8.1 Spectrum (h)
Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature, called the Planck spectrum or Planck's law. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum.
As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark, the first faint glow appears as a "ghostly" grey. With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises. When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The Sun, with an effective temperature of approximately 5800 K, is an approximately black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well.
Black-body radiation provides insight into the thermodynamic equilibrium state of cavity radiation. If each Fourier mode of the equilibrium radiation in an otherwise empty cavity with perfectly reflective walls is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes this implies infinite heat capacity (infinite energy at any non-zero temperature), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe. Instead, in quantum theory the occupation numbers of the modes are quantized, cutting off the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.
7.8.2 Equations of blackbody radiation (h)
7.8.2.1 Planck's law of black-body radiation (sh)
Planck's law states that:
Where, I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T;
h is the Planck constant;
c is the speed of light in a vacuum;
k is the Boltzmann constant;
ν is the frequency of the electromagnetic radiation; and
T is the absolute temperature of the body.
7.8.2.2 Wien's displacement law (sh)
Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency. A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body is at a maximum, �max, is a function only of the temperature
�max = b/T
where the constant, b, known as Wien's displacement constant, is equal to 2.8977721(26)×10−3 K m.
7.8.2.3 Stefan–Boltzmann law (sh)
The Stefan–Boltzmann law states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:
J* = σ T4
where j*is the total power radiated per unit area, T is the absolute temperature and σ = 5.67×10−8 W m−2 K−4 is the Stefan–Boltzmann constant.
7.9 Emissitivity (mh)
The emissivity of a material (usually written ε or e) is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature. A true black body would have an ε = 1 while any real object would have ε < 1. Emissivity is a dimensionless quantity.
Emissivity depends on factors such as temperature, emission angle, and wavelength. A typical physical assumption is that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the "gray body assumption".
Although it is common to discuss the "emissivity of a material" (such as the emissivity of highly polished silver), the emissivity of a material does in general depend on its thickness. The emissivities quoted for materials are for samples of infinite thickness (which, in practice, means samples which are optically thick) — thinner samples of material will have reduced emissivity.
When dealing with non-black surfaces, the deviations from ideal black body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoff's law of thermal radiation: emissivity equals absorptivity (for an object in thermal equilibrium), so that an object that does not absorb all incident light will also emit less radiation than an ideal black body.
Most emissivities found in handbooks and on websites of many infrared imaging and temperature sensor companies are the type discussed here, total emissivity. However, the distinction needs to be made that the wavelength-dependent or spectral emissivity is the more significant parameter to be used when one is seeking an emissivity correction for a temperature measurement device.
7.9.1 Emissivity of Earth’s Atmosphere (h)
The emissivity of Earth's atmosphere varies according to cloud cover and the concentration of gases that absorb and emit energy in the thermal infrared (i.e., wavelengths around 8 to 14 micrometres). These gases are often called greenhouse gases, from their role in the greenhouse effect. The main naturally-occurring greenhouse gases are water vapor, carbon dioxide, methane, and ozone. The major constituents of the atmosphere, N2 and O2, do not absorb or emit in the thermal infrared.
7.9.2 Emissivity between two Walls (h)
Given two parallel walls whose facing surfaces have respective emissivities and at a given wavelength, a certain fraction of the radiation of that wavelength just inside one wall will leave that wall and enter the other. By Kirchhoff's law of thermal radiation for a given wavelength, whatever portion of the radiation incident on a surface, from either side, that does not pass
through the surface as emission to the other side, is reflected. When this reflected radiation is neglected, the proportion of radiation emitted from the first wall is , and the proportion of that entering the second wall is therefore .
When reflection is taken into account, what does not enter the second wall is reflected back to
the first wall, initially in an amount . A fraction of this is then reflected back to the second wall, thereby augmenting the original emission from the first wall. These reflections bounce back and forth in diminishing quantity. Solving for the steady state then gives as the total proportion of radiation entering the second wall
This formula is symmetric, and the proportion of radiation just inside the second wall that enters the first wall is the same. This is true regardless of what reflections and absorptions take place inside the two walls away from their facing surfaces, since the formula only concerns the radiation leaving one wall for the other.
The quantities in these formulas are intensities rather than amplitudes, the appropriate choice when the walls are many wavelengths apart as the reflected and transmitted beams will then combine incoherently. When the walls are only a few wavelengths apart, as arises for example with the thin films used in the manufacture of optical coatings, the reflections tend to combine coherently, resulting in interference. In such a situation the above formula becomes invalid, and one must then add amplitudes instead of intensities, taking into account the phase shift as the gap is traversed and the phase reversal that occurs with reflection, concerns that did not arise in the incoherent large-gap or thick-film case.
7.10 Absorptivity (mh)
In spectroscopy, the absorbance (also called optical density of a material is a logarithmic ratio of the radiation falling upon a material, to the radiation transmitted through a material.Absorbance is a quantitative measure expressed as a logarithmic ratio between the radiation falling upon a material and the radiation transmitted through a material.
where is the absorbance at a certain wavelength of light , I1 is the intensity of the radiation (light) that has passed through the material (transmitted radiation), and I0 is the intensity of the radiation before it passes through the material (incident radiation).
The term absorption refers to the physical process of absorbing light, while absorbance refers to the mathematical quantity. Also, absorbance does not always measure absorption: if a given sample is, for example, a dispersion, part of the incident light will in fact be scattered by the dispersed particles, and not really absorbed. However, in such cases, it is recommended that the term "attenuance" be used, which accounts for losses due to scattering and luminescence.
7.11 Stefan’s Boltzmann Law (mh)
The Stefan–Boltzmann law, also known as Stefan's law, is a relation which described the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power), j* is directly proportional to the fourth power of the black body's thermodynamic temperature T:
The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other known constants of nature. The value of the constant is
where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. Thus at 100 K the energy flux density is 5.67 W/m2, at 1000 K 56,700 W/m2,
Derivation-
The fact that the energy density of the box containing radiation is proportional to T4 can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure p is related to the internal energy density u:
From the fundamental thermodynamic relation
dU = T dS - p dV ,
we obtain the following expression, after dividing by dV and fixing T:
The last equality comes from the following Maxwell relation:
From the definition of energy density it follows that
and
Now, the equality
after substitution of and for the corresponding expressions, can be written as
Since the partial derivative can be expressed as a relationship between only and (if one isolates them on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes
,
which leads immeadiately to , with as some constant of integration.
7.12 KIRCHOFF’S LAW (mh)
In thermodynamics, Kirchhoff's law of thermal radiation refers to wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium, including radiative exchange equilibrium.
A body at temperature T radiates electromagnetic energy. A perfect black body in thermodynamic equilibrium absorbs all light that strikes it, and radiates energy according to a unique law of radiative emissive power for temperature T, universal for all perfect black bodies. Kirchhoff's law states that:
For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power.
Here, the dimensionless coefficient of absorption (or the absorptivity) is the fraction of incident light (power) that is absorbed by the body when it is radiating and absorbing in thermodynamic equilibrium. In slightly different terms, the emissive power of an arbitrary opaque body of fixed size and shape at a definite temperature can be described by a dimensionless ratio, sometimes called the emissivity, the ratio of the emissive power of the body to the emissive power of a black body of the same size and shape at the same fixed temperature. With this definition, a corollary of Kirchhoff's law is that for an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity. In some cases, emissive power and absorptivity may be defined to depend on angle, as described below.
Kirchhoff's Law has another corollary: the emissivity cannot exceed one (because the absorptivity cannot, by conservation of energy), so it is not possible to thermally radiate more energy than a black body, at equilibrium. In negative luminescence the angle and wavelength integrated absorption exceeds the material's emission, however, such systems are powered by an external source and are therefore not in thermodynamic equilibrium.
Before Kirchhoff's law was recognized, it had been experimentally established that a good absorber is a good emitter, and a poor absorber is a poor emitter. Naturally, a good reflector must be a poor absorber. This is why, for example, lightweight emergency thermal blankets are based on reflective metallic coatings: they lose little heat by radiation.
7.12.1 Theory (h)
In a blackbody enclosure that contains electromagnetic radiation with a certain amount of energy at thermodynamic equilibrium, this "photon gas" will have a Planck distribution of energies.One may suppose a second system, a cavity with walls that are opaque, rigid, and not perfectly
reflective to any wavelength, to be brought into connection, through an optical filter, with the blackbody enclosure, both at the same temperature. Radiation can pass from one system to the other. For example, suppose in the second system, the density of photons at narrow frequency band around wavelength \lambda were higher than that of the first system. If the optical filter passed only that frequency band, then there would be a net transfer of photons, and their energy, from the second system to the first. This is in violation of the second law of thermodynamics, which requires that there can be no net transfer of heat between two bodies at the same temperature.In the second system, therefore, at each frequency, the walls must absorb and emit energy in such a way as to maintain the black body distribution.The absorptivity �α is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength.The emissivity of the wall is defined as the ratio of emitted energy to the amount that would be radiated if the wall were a perfect black body.These two quantities must be equal, or else the distribution of photon energies in the cavity will deviate from that of a black body. This yields Kirchhoff's law:
�α =��
By a similar, but more complicated argument, it can be shown that, since black body radiation is equal in every direction (isotropic), the emissivity and the absorptivity, if they happen to be dependent on direction, must again be equal for any given direction.Average and overall absorptivity and emissivity data are often given for materials with values which differ from each other. For example, white paint is quoted as having an absorptivity of 0.16, while having an emissivity of 0.93.This is because the absorptivity is averaged with weighting for the solar spectrum, while the emissivity is weighted for the emission of the paint itself at normal ambient temperatures.
Example- Each square metre of the sun’s surface radiates energy at the rate of 6.3 × 107 W m–2. Assuming that Stefan’s law applies to the radiation, calculate the temperature of the sun’s surface. Given Stefan’s constant s = 5.7 × 10–8 W m–2 K–4. Sol: Given Q = 6.3 × 1078 W m–2
σ = 5.7 × 10–8 W m–2 K–4
T = ?
Now, Q = σT4 T = (Q/σ)1/4 = (6.3 × 107/5.7 × 10–8)1/4 = 5765 K
The temperature of sun’s surface is 5765 K.
Example- The tungsten filament of an electric lamp has a length of 0.25 m and a diameter 6 × 10–5 m. The power rating of the lamp is 100 W. If the emissivity of the filament is 0.8, estimate the steady temperature of the filament. Stefan’s constant = 5.67 × 10–8 W m–2 K–4.
Sol: Area of the filament = 2p × (radius) × (length)
A = 2p × (3 × 10–5) × 0.25 = 4.71 × 10–5 m2
Now Q = σεT4, where Q is the energy radiated per second per unit area at absolute temperature T. Therefore, the energy radiated per second (or power radiated) from the filament of area A is, P = εσAT4.
When the temperature is steady, power radiated from filament = power received = 100 W
AεσT4 = 100 W
Now A = 4.71 × 10–5 m2, ε = 0.8 and σ = 5.67 × 10–8 W m–2 K-4
Substituting these values, we have,
T = (100/4.71 × 10–5 × 0.8 × 5.67 ×10–8)1/4 = 2616 K
Example- A 100 Watt bulb is having length 40 cm, radius 0.05 m. if emissivity is 0.85. Calculate the temperature?
Sol. Given that: length l = 0.4 m,
radius r = 0.05 m,
emissivity= 0.85,
T =?
Area A = Πr2= 3.142 (0.05)2= 0.007855 m2
Using the formula
P = �σAT4
T4 = 0.85 /5.67 × 10-8 x 0.007855
Sol. T = 716.9 K.
REVIEW QUESTIONS
1. A block of shiny silver (absorptance = 0.23) has a bubble inside it of radius 2.2 cm, and it is held at a temperature of 1200 K.
2. By what factor should the temperature of a black body be increased so that a) The integrated radiance (over all frequencies) is doubled?
b) The frequency at which its radiance is greatest is doubled?
c) The spectral radiance per unit wavelength interval at its wavelength of maximum spectral radiance is doubled?
3. A hot black body of mass 64 gm, area 16 sq cm and specific heat 0.1 cal/gm/ °C is allowed to cool inside an evacuated enclosure surrounded by melting ice. It is found that at 300 °C, the body cools at the rate of 21 °C per minute; calculate Stefan's constant.
4. A solid copper sphere cools at the rate of 2.8 degree per minute when its temperature is 127 °C. At what rate will a solid copper sphere of twice the radius cool when its temperature is 227 °C, if in both cases the surroundings are maintained at 27 °C and the conditions are such that Stefan's law may be applied.
5. Calculate the maximum net rate of loss of heat by radiation from a sphere of 10 cm radius at a temperature of 200 °C when the surroundings are at a temperature of 20 °C, if Stefan's constant is 5.7 × 10−5 ergs/sq cm/s/deg4.
6. Heat is supplied to a slab of compressed cork 5 cm thick and of effective area 2 sq m by a heating coil spread over its surface. When the current in this coil is 1.18 amperes and the potential difference across its ends 20 volts, the steady temperatures of the faces of the slab are 12.5°C and 0°C. Assuming that the whole of the heat developed in the coil is conducted through the slab, calculate the thermal conductivity of the cork.
7. A block of iron (specific heat: 0.45 kJ/kg.K) with a mass of 20 kg is heated from its initial temperature of 10⁰C to a final temperature of 200⁰C by keeping it in thermal contact with a thermal energy reservoir (TER) at 300⁰C. All other faces of the block are completely insulated. Determine the amount of (a) heat and (b) entropy transfer from the reservoir. (c) Also calculate the entropy generated in the system's universe due to this heat transfer.
8. A 0.4 m3 rigid tank contains refrigerant-134a initially at 250 kPa and 45 percent quality. Heat is transferred now to the refrigerant from a source at 37⁰C until pressure rises to 420 kPa. Determine (a) the entropy change of the refrigerant, (b) the entropy change of the heat source, and (c) the total entropy generation in the universe due to the process.
9. A frictionless piston is used to provide a constant pressure of 500 kPa in a cylinder containing steam originally at 250⁰C with a volume of 3m3. Determine (a) the final temperature if 3000 kJ of heat is added, (b) the work done by piston.
10. Nitrogen at an initial state of 80oF, 25 psia and 6ft3 is pressed slowly in an isothermal process to a final pressure of 110 psia. Determine (a) the work done during the process. (b) What-if scenario:How would the answer change if pressure was 100 psia?
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