1

1. PHYSICAL QUANTITIES & MEASUREMENTS

1.1 INTRODUCTION TO PHYSICS

a. Definition of Physics

Physics, the most fundamental science, is concerned with the basic principles of the Universe. It is one of the foundations on which the other physical sciences astronomy, chemistry, and geology are based.

Physics is the study of the properties and nature of matter, the different forms of energy and the ways in which matter and energy interact in the world around us. To understand physics, we need to know Model, Theories and Laws in describing a phenomenon.

The study of the laws that determines the structure of the universe with reference to the matter and energy of which it consists. It is not concerned not with chemical changes that occur but with the forces that exist between objects and the interrelationship between matter and energy.

b. The goal of Physics

The goal of Physics is to provide an understanding of nature by developing theories based on experiments

c. The beauty of Physics

The beauty of Physics lies in the simplicity of its fundamental theories and in the way just a small number of basics concepts, equations, and assumptions can alter and expand our view of the world.

d. The areas of Physics

Physics can be divided into five areas:

i. Mechanics :- which is concerned with the effects of forces on material objects. Also covers the main concepts of physics, e.g. forces, energy and the properties of matter.

ii. Thermodynamics :- which deals with heat, temperature, and the behavior of large numbers of particles. Also explains heat energy in terms of its measurement and the effects of its presence and transference. Includes the gas laws.

iii. Electromagnetism :- which deals with charges, currents, and electromagnetic fields. Also explains the forms, uses and characteristics of these two linked phenomena.

iv. Relativity (Year : 1905) :- a theory that describes particles moving at any speed, and connects space and time

v. Quantum Mechanics (Year: 1900) :- a theory dealing with behavior of particles at the submicrospic level as well as the macroscopic world

Since the turn of the century, however, quantum mechanics and relativistic physics have become increasingly important; the growth of modern physics has been accompanied by the studies of atomic physics, nuclear physics (1896) and particle physics.

e. Some terms of Physics

i. Astrophysics : - The Physics of astronomical bodies and their interactions. Astrophysics also studies the physical and chemical processes involving astronomical phenomena. Astrophysics deals with stellar structure and evolution (including the generation and transport of energy within stars), the properties of the interstellar medium and its interactions with stellar systems, and the structure and dynamics of systems of stars and systems of galaxies.

ii. Geophysics : - The branch of science in which the principles of mathematics and physics are applied to the study the earths crust and interior. It includes the study of earthquake waves, geomagnetism, gravitational fields, and electrical conductivity using precise quantitative principles. In applied geophysics the techniques are applied to the discovery and location of economic minerals (e.g. petroleum).

iii Biophysics :- The study of the physical aspects of biology.

iv. Theoretical Physics : - The study of physics by formulating and analyzing theories that describe natural processes. Theoretical physics is complementary to the study of physics by experiment.

v. Experimental Physics : - The study of physics by experiment

vi. Mathematical Physics :- The branch of theoretical physics concerned with the mathematical aspects of theories in physics

vii. Thinking Physics : - The study of Physics which emphasis more on critical thinking and teaching physical concepts

viii. Laws :- A law is a descriptive principle of nature that holds in all circumstances covered by the wording of the law. Some laws are named after their discoverers (e.g. * Boyles law); some laws, however, are known by their subject matter to describe them (e.g. * the law of conservation of mass), while other laws use both the name of the discoverer and the subject matter to describe them (e.g. * Newtons law of gravitation).

ix. Theory :- A description of nature that encompasses more than one law but has not achieved the uncontrovertibly status of a law . Theories are often both eponymous and descriptive of the subject matter (e.g. Einsteins theory of relativity and Darwins theory of evolution).

x. Hypothesis :- A theory or law that retains the suggestion that it may not be universally true. Some hypothesis about which no doubt still linger have remained hypotheses ( e.g. Avogadros hypothesis ) for no clear reason.

1.2 QUANTITIES & UNITS

All things in classical mechanics can be expressed in terms of the fundamental dimension or unit:

Dimension

Unit

Length

L meter

Mass

M

kilogram

Time

T

second

For example:

Speed has dimension of L / T (i.e. Km per hour).

Force has dimension of ML / T2 etc...Length:

Distance

Length (m)

Radius of visible universe

1 x 1026To Andromeda Galaxy

2 x 1022To nearest star

4 x 1016Earth to Sun

1.5 x 1011Radius of Earth

6.4 x 106Sears Tower

4.5 x 102Football field

1.0 x 102Tall person

2 x 100Thickness of paper

1 x 10-4Wavelength of blue light

4 x 10-7Diameter of hydrogen atom

1 x 10-10Diameter of proton

1 x 10-15Time:

Interval

Time (s)

Age of universe

5 x 1017Age of Grand Canyon

3 x 101432 years

1 x 109One year

3.2 x 107One hour

3.6 x 103Light travel from Earth to Moon1.3 x 100One cycle of guitar a string

2 x 10-3One cycle of FM radio wave 6 x 10-8Lifetime of neutral pi meson

1 x 10-16

Lifetime of top quark

4 x 10-25Mass:

Object

Mass (kg)

Milky Way Galaxy

4 x 1041Sun

2 x 1030Earth

6 x 1024Boeing 747

4 x 105Car

1 x 103Student

7 x 101Dust particle

1 x 10-9Top quark

3 x 10-25

Proton

2 x 10-27Electron

9 x 10-31

Neutrino

1 x 10-38Units...

SI (System International) Units:

mks: L = meters (m), M = kilograms (kg), T = seconds (s)

cgs: L = centimeters (cm), M = grams (gm), T = seconds (s)

Derived Units :

Newton, Joule, Watt, Ohm . and etc.

British Units: Inches, feet, miles, pounds, slugs...

fps : L = foot, M = pound, T = second

We will use mostly SI units with mks system, but you may run across some problems (rarely happen) using British units. You should know how to convert back & forth.

The 7 International System of Units (SI)

Derived Units QuantityUnitAbbreviationIn terms of Base Units

ForceNewtonNkg ms-2

Energy & WorkJouleJkg m2s-2

PowerWattWkg m2s-3

PressurePascalPakg / (ms2 )

Electric ChargeCoulombCA s

Electric PotentialVoltVkg m2 / (A s3 )

CapacitanceFaradFA2 s4 / (kg m2 )

InductanceHenryHkg m2 / (s2 A2 )

Magnetic FluxWeberWbkg m2 / (A s2 )

Standard Prefixes : used to denote multiple of ten

FactorPrefixSymbolFactorPrefixSymbol

10-1decid101dekada

10-2centic102hectoh

10-3millim103kilok

10-6microm106MegaM

10-9nanon109GigaG

10-12picop1012TeraT

10-15femtof1015PetaP

10-18atoa

1.3 DIMENSIONAL ANALYSIS

DIMENSIONS

Many physical quantities can be expressed in terms of a combination of fundamental dimensions such as

[Length]

L

[Time]

T

[Mass]

M

[Current]

A

[Temperature] [Amount] N

The symbol [ ] means dimension or stands for dimension

There are physical quantities which are dimensionless:

numerical value

ratio between the same quantity angle

some of the known constants like ln, log and etc.

Dimensional Analysis

Dimension analysis can be used to:

Derive an equation.

Check whether an equation is dimensionally correct. However, if an equation is dimensionally correct, it doesnt necessarily mean it is correct. Find out dimension or units of derived quantities.

Derived an Equation (Quantities)Example 1:

Velocity = displacement / time

[velocity] = [displacement] / [time]

= L / T

= LT-1

v = s / t

Example 2: The period of a pendulum

The period P of a swinging pendulum depends only on the length of the pendulum l and the acceleration of gravity g.What are the dimensions of the variables?t T

m M

L

g LT-2

Write a general equation:

By using the dimension method, an expression could derived that relates T, l and g

T ma bgcwhereby a, b and c are dimensionless constantsThusT = kma bgcWrite out the dimensions of the variables

[T] = [ma][ b][gc]

= MaLb(LT-2)c

= MaLbLcT-2c

T1 = MaL b+c T-2c

Using indices

a = 0

-2c = 1 c = -

b + c = 0

b = -c = T = kma bgcT = km0 g-

Whereby, the value of k is known by experiment

ExercisesThe viscosity force, F going against the movement of a sphere immersed in a fluid depends on the radius of the sphere, a, the speed of the sphere, v and the viscosity of the fluid, . By using the dimension method, derive an equation that relates F with a, v and .

(given that )

To check whether a specific formula or an equation is homogenous

Example 1

s= vt[s] = [v] [t]L.H.S [s] = L

R.H.S [v] [t] = LT-1T

= L

Thus, the left hand side = right hand side, rendering the equation as homogenous

Example 2

Given that the speed for the wave of a rope is , check its homogeneity by using the dimensional analysis.

L.H.S. [ C ] = [ LT-1]2 = L2T-2R.H.S. [ C ] = MLT-2, [ m ] =M

Conclusion: The above equation is not homogenous (L.H.S R.H.S)

Exercises

Show that the equations below are either homogenous or otherwise

v = u + 2as

s = ut + at2

Find out dimension or units of derived quantities

ExampleConsider the equation , where m is the mass and T is a time, therefore dimension of k can be describe as

= MT-2 unit: kgs-2

thus, the units of k is in kgs-2

Exercise

The speed of a sound wave, v going through an elastic matter depends on the density of the elastic matter, and a constant E given as equation v= E - -. Determine the dimension for E in its SI units

Dimensional Analysis

Example:

The period P of a swinging pendulum depends only on the length of the pendulum l and the acceleration of gravity g. Which of the following formulas for P is correct?Given: l has units of length (L) and g has units of (L / T 2).

a) P= 2((lg)2 (b) (c)

Realize that the left hand side P has units of time (T )Solution:(a) Not Right!

(b) Not Right!(c)

This has the correct unit! This must be the answer!

1.4 SCALAR AND VECTOR

Scalars:

Scalars are quantities which have magnitude without direction

Examples of scalars

Mass

Temperature

Kinetic energy

Time

Amount

Density

chargeVector:

A vector is a quantity that has both magnitude (size) and direction

It is represented by an arrow whereby

the length of the arrow is the magnitude, and

the arrow itself indicates the direction

The symbol for a vector is a letter

with an arrow over it

Two ways to specify

It is either given by

a magnitude A, and

a direction (

Or it is given in the x and y components as

Ax

Ay

Ax = A cos (

Ay = A sin (

The magnitude (length) of A is found by using the Pythagorean Theorem

The length of a vector clearly does not depend on its direction.

The direction of A can be stated as

Some Properties of Vectors:

Equality of Two Vectors

Two vectors A and B may be defined to be equal if they have the same magnitudes and point in the same directions. i.e. A= B

Negative of a vector:

The negative of vector A is defined as giving the vector sum of zero value when added to A . That is, A + (- A) = 0. The vector A and A have the same magnitude but are in opposite directions.

Scalar Multiplication:

The multiplication of a vector A by a scalar (

- will result in a vector B

- the magnitude is changed but not the direction

Do flip the direction if ( is negativeIf ( = 0, therefore B = (, A = 0, which is also known as a zero vector

(((A) = ((A = (((A)

((+()A = (A + (AVector Addition

The addition of two vectors, A and B- will result in a third vector, C called the resultant

C = A + B

Geometrically (triangle method of addition)

put the tail-end of B at the top-end of A C connects the tail-end of A to the top-end of B We can arrange the vectors as we like, as long as we maintain their length and direction

More than two vectors?

Vector Subtraction:

It is equivalent to adding the negative vectors.

Rules of Vector Addition

Commutative

Associative

Distributive

Parallelogram method of addition (tail-to-tail)

The magnitude of the resultant depends on the relative directions of the vectors

Unit Vectors

A Vector whose magnitude is 1 and dimensionless.

The magnitude of each unit vector equals a unity, that is

Useful examples for the Cartesian unit vectors [ i, j, k ]

- they point in the direction of the x, y and z axes respectively

Component of a vector in 2-D:

A vector A can be resolved into two components

Ax and Ay

The component of A are:Ax = Ax = A cos Ay = Ay = A sin The magnitude of A:

The direction of A:

The unit vector notation for the vector A is written

A = Axi + Ayj

Component of vector in 3-D:

A vector A can be resolved into three components Ax , Ay and Az.A = Axi + Ayj + Azk

If

Dot Product ( Scalar ) of two vectors:

If = 900 (normal vectors) then the dot product is zero. |A B| = AB cos 90 = 0 and i j = j k = i k = 0

if = 00 (parallel vectors) it gets its maximum value of 1

|A B| = AB cos 0 = 1 and i j = j k = i k = 1

The dot product is commutative.

Use the distributive law to evaluate the dot product if the components are known.

Cross product ( vector) of two vectors:

The magnitude of the cross product is given by

The vector product creates a new vector. This vector is normal to the plane defined by the original vectors and its direction is found by using the right hand rule.

If = 00 (parallel vectors) then the cross product is zero.

If = 900 (normal vectors) it gets its maximum value. The relationship between vectors i , j and k can be described as

i x j = - j x i = k

j x k = - k x j = i

k x i = - i x k = j

The resultant has a magnitude A + B when A is oriented in the same direction as B. The resultant vector A + B = 0 when A is oriented in the direction opposite to B, and when A = B

No. The magnitude of a vector A is equal to (Ax2 + Ay2 +Az2).Therefore, if any component is nonzero, A cannot be zero. Proof of this generalization of the Pythagorean theorem.

A = -B, therefore the components of the two vectors must have opposite sings and equal magnitudes.

Tan = Ry / Rx

= (Ax + By) / (Ax + Bx)

1.5 MEASUREMENTS AND ERRORSTerminology: True value standard or reference of known value or a theoretical value

Accuracy closeness to the true value

Precision reproducibility or agreement with each other for multiple trials

Types of Errors

i. Systematic errors Sometimes called bias due to error in one direction- high or low

Known cause

Operator

Wrong calibration of glassware, sensor, or instrument

When it is determined, it can be corrected

May be of a constant or proportional nature

ii. Random errors Cannot be determined (no control over)

Random nature causes both high and low values which will average out

Multiple trials help to minimize

Accuracy and Precision

The Uncertainty:Example:

True value of thickness of a book is 5cm.

Student A uses meter ruler and measures the thickness to be 4.9cm with an uncertainty of 0.1cm.

Student B , with Vernier caliper, found it to be 4.85cm with an uncertainty of 0.01cm.

We may say,

Student A has more accurate value, but less precise.

Student B got a more precise value, but less accurate (due to the faulty caliper. Un- calibrate !)

However, after sending the caliper to be calibrated, student B performs the measurement again and found the thickness is 4.98cm. So, now he has more accurate and more precise value.

Note: We always report a measurement in a way that would includes the uncertainty carried by the instrument.

Combining uncertainties + and -

Adding or subtracting quantities then sum all individual absolute uncertainties

2.1 0.1 + 2.0 0.2 = 4.1 0.3

2.1 0.1 - 2.0 0.2 = 0.1 0.3

This method overestimates the final uncertainty.Combining uncertainties x and When Dividing or multiplying quantities, then sum all of the individual relative uncertainties

(2.5 0.1) x (5.0 0.1)

= (2.5 4%) x (5.1 2%) =12.5 6% (or 0.75 or 0.7)

(21 6%) / (5.0 4%)

= 4.12 10% or 4.2 0.42 or 4.2 0.4

However it will overestimate final uncertainty.The Significant figures

The number of Significant figures of a numerical quantity is the number of reliably known digits it contains.

For measured quantity, it is defined as all of the digits that can be read directly from the instrument used in making the measurement plus one uncertain digit that is obtained by estimating the fraction of the smallest division of the instruments scale.

Note: Exact quantities are considered as having unlimited number of significant figures. We need to be concerned with significant figures only when dealing with measurements that have required some estimation.

For example,

Reading of the thickness of a book is

5.0cm or 50mm from meter ruler (with 2 sf)

5.00cm or 50.0 mm from vernier caliper. (with 3 sf)

The rules of significant figures:

1. Any figure that is non-zero, is considered as a significant figure.2. Zeros at the beginning of a number are not significant

Example: 0.254 ----------------- 3 s.f

3. Zeros within a number are significant.

Example: 104.6 m ---------------- 4 s.f

4. Zeros at the end of a number after the decimal point are significant.

Example: 27050.0 ------------------- 6 s.f

5. Zeros at the end of a whole number without a decimal point may or may not be significant.

It depends on how that particular number was obtained, using what kind of instrument, and

the uncertainty involved.

Example: 500m ------------------- could be 1 or 3 sf.

Convert the unit:

500m = 0.5km ( 1 sf )

500m = 50 000cm ( 5 sf )

Addition and Subtraction processes

The rule:

The final result of an addition and/or subtraction should have the same number of significant figures as the quantity with the least number of decimal places used in the calculation.

Example:

23.1 + 45 + 0.68 + 100 = 169

Example:

23.5 + 0.567 + 0.85 = 24.9

Multiplication and division processesThe rule:

The final result of an multiplication and/or division should have the same number of significant figures as the quantity with the least number of significant figures used in the calculation.

Example:

0.586 x 3.4 = 1.9924

= 2.0

Example:

13.90 / 0.580 = 23.9655 = 24.0

Estimating the slope

1. Simple conservative method

Plot error bars on the graph

Draw maximum (mmax) and minimum (mmin) slopes.

The simplest method is to plot the data (and associated error bars) and draw 2 lines through the points. One with a maxiumum slope that still manages to go through all of the error bars and one with the minimum slope that does likewise.

The average slope and uncertainty on the slope are given above.

This is a very simple method but usually overestimates the uncertainty especially if the data is reasonably linear to start with and the error bars are large.

Quantity

SI Units

Symbol

Length

meter

m

Mass

kilogram

Kg

Time

second

s

Electric current

Ampere

I

Temperature

kelvin

K

Luminous Intensity

candela

cd

Amount of Substance

mol

mol

Grab the whole picture !

A

Ay

Ax

x

y

A

A

A

(

x

y

A

EMBED Equation.DSMT4

Dimension

Analysis

EMBED Equation.DSMT4

Significant Figures

Accuracy & Uncertainty

Scalar Quantities

Vector Quantities

Instruments

Units

Quantities

Measurements

B

A

A

B

B

A

-A

B = ( A

A

B

C

x1

x5

x4

x3

x2

(xi

(xi = x1 + x2 + x3 + x4 + x5

A

-B

A - B

B

A - B

C =

A + (-B)

C =

A + B = B + A

A

B

A + B

B

A

A + B

C= 48.2 km

_1138525158.unknown

_1138526026.unknown

_1138526906.unknown

_1138536093.unknown

_1138543770.unknown

_1138543818.unknown

_1138543639.unknown

_1138535899.unknown

_1138526318.unknown

_1138526610.unknown

_1138526064.unknown

_1138525325.unknown

_1138525402.unknown

_1138525177.unknown

_1138524326.unknown

_1138524708.unknown

_1138525115.unknown

_1138524367.unknown

_1138523613.unknown

_1138524240.unknown

_1138523327.unknown

_1138523337.unknown

_1138523315.unknown