Chapter 1 Physical Quantities andUnitsIntroductionWhat is physic ? Definition of physics derives from Greek word means nature. Each theory in physics involves:(a) Concept of physical quantities.(b) Assumption to obtain mathematical model.(c) Relationship between physical concepts. directly proportional linearly proportional exponentially proportional(d) Procedures to relate mathematical models to actual measurements from experiments.(e) Experimental proofs to devise explanation to nature phenomena.

1.1 Basic Quantities and International System of Units (SI units)> Physical quantityA physical quantity is a quantity that can be measured. Physical quantity consist of anumerical magnitude and a unit.Example:250 ml (magnitude and unit)> Basic quantityQuantity that cannot be derived from other quantities. This quantity is important because it can be easily produced does not change its magnitude is internationally accepted> SI unitsThe unit of a physical quantity is the standard size used to compare different magnitudesof the same physical quantity.> Systems of unitsSeveral systems of units have been in use. Example: The MKS (meter-kilogram-second) system The cgs (centimeter-gram-second) system British engineering system: foot for length, pound for mass and second for time.Today the most important system of unit is the Systems International or Sl units.

Basic Quantity and the SI Base Units Physical quantities can be divided into two categories:1. basic quantities and2. derived quantities.The corresponding units for these quantities are called base units and derived units.

Basic Quantities In the interest of simplicity,seven basics quantities1, consistent with a full description of thephysical world, have been chosen.Basic quantitySymbolDimension(base quantity symbol)Definition2SI units

LengthlLlengthmost commonly refers to the longest dimension of an objectMeter

MassmMMass, more specificallyinertial mass, can be defined as a quantitative measure of an objects resistance to accelerationKilogram

TimetTTimeis a dimension in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between themSecond

ElectriccurrentIAElectric currentis a flow of electric charge through a conductive mediumAmpere

ThermodynamictemperatureTqTemperatureis a physical property of matter that quantitatively expresses the common notions of hot and cold.Kelvin

Amount of substances, Quantity ofmatternNAmount of substanceis a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particlesMole

Luminous intensityIvJluminous intensityis a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angleCandela

Base UnitsThere are onlyseven base unit3in SI system.SI Base unitsSymbolDefinition1

Metrem"The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second."17th CGPM (1983, Resolution 1, CR, 97)

Kilogramkg"The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram."3rd CGPM (1901, CR, 70)

Seconds"The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."13th CGPM (1967/68, Resolution 1; CR, 103)"This definition refers to a caesium atom at rest at a temperature of 0 K."(Added by CIPM in 1997)

AmpereA"The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 107newton per metre of length."9th CGPM (1948)

KelvinK"The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water."13th CGPM (1967/68, Resolution 4; CR, 104)"This definition refers to water having the isotopic composition defined exactly by the following amount of substance ratios: 0.000 155 76 mole of2H per mole of1H, 0.000 379 9 mole of17O per mole of16O, and 0.002 005 2 mole of18O per mole of16O."(Added by CIPM in 2005)

MoleMol"1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is mol.2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles."14th CGPM (1971, Resolution 3; CR, 78)"In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to."(Added by CIPM in 1980)

Candelacd"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 1012hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."16th CGPM (1979, Resolution 3; CR, 100)

PrefixesFor very large or very small numbers, we can use standard prefixes with the base units.Prefixteragigamegakilodecicentimilimicronanopico

Factor10210910610310-110-210-310-610-910-12

SymbolTGMkdcmnP

Example 1.1

Derived quantities and derived unitsDerived quantityQuantity that derived from basic quantities through multiplication and division.For example,Derived quantityDerive from base quntity ofDerived unit

Arealength x lengthm2

Volumelength x length x lengthm3

Densitykg m-3

Velocitym s-1

Accelerationm s-2

Frequencys-1/hz

MomentumMass x velocityKg ms-1

ForceMass x accelerationKg ms-2

PressureN m-2

EnergyKg m2s-2

The derived unit changeExample 1.2

1.2 Dimensions(Dimensi)and Physical QuantitiesThe dimension of a physical quantities is the relation between the physical quantity and thebase quantities.[ ]The dimension of (pronounce its loudly) or the base quantity ofExample [v] the dimension of velocity , this means that the base quantities in the velocity.Example 1.3

Use of dimensionsTo check the homogeneity of physical equationsConcept of homogeneous

The dimensions on both sides of an equation are the same.Those equations which are not homogeneous are definitely wrong.However, the homogeneous equation could be wrong due to the incomplete or has extra terms.The validity of a physical equation can only be confirmed experimentally.In experiment, graphs have to be drawn then. A straight line graph shows the correct equation and the non linear graph is not the correct equation.Deriving a physical equationAn equation can be derived to relate a physical quantity to the variables that the quantity depends on.Example 1.4Derivation of Physical EquationFrom observations and experiments, a physical quantity may be found to be dependent on a few other physical quantity. To find this relationship we use dimension method.Example 1.5Example 1.6

1.3 Scalar and Vectors> A scalar quantity is a physical quantity which has only magnitude. For example, mass, speed , density, pressure, .> A vector quantity is a physical quantity which has magnitude and direction. For example, force, momentum, velocity , acceleration .In most cases in physic, the physic quantity is express in vector. If the number(magnitude) can be operated through Subtract, Add, multiplication and fraction. Then the vector also can be threat the same way except fraction, but its have to follow the rule that govern them.

Graphical representation of vectorsA vector can be represented by a straight arrow,

The length of the arrow represents the magnitude of the vector.The vector points in the direction of the arrow.Basic principle of vectors Two vectors P and Q are equal if:a) Magnitude of P = magnitude of Q(b) Direction of P = direction of Q When a vector P is multiplied by a scalar k, the product is k P and the direction remains the same as P.The vector -P has same magnitude with P but comes in the opposite direction.

Principles of vectors

(a) Substitute of Vector (Relative of)Relative velocityLet us look at two cases: VA= 10 ms-1(faster) VB= 3 ms-1. (slower)Case oneThe velocity ofA relative to B= (VA VB) (comparing faster toward slower)= (10- 3) ms=7 ms-1(in forward direction).(mean that A is 7 ms-1faster than B)Case twoThe velocity ofB relative to A= (VB VA)= (3 10) ms= -7 ms-1(in backwards direction).We observe that(VB VA) and (VA VB) are same magnitude but different direction.(b) Sum of vectors (Resultant of)If there are two or more vector , these vector can be add to form a single vector called aResultantvector.To solve the problem involving vectors in two dimension, we usually used any one of these method depend on the information given.

Method 1: Parallelogram of vectorsIts the drawing method. The drawing of the parallelogram need to be draw according scale and angle given in the question. The instrument used for this drawing are:(a) ruler(b) protractor(c) sharp pencilIt two vectorsandare represented in magnitude and direction by the adjacent sides OA and OB of a parallelogram OABC, then OC represents their resultant.

This method is used when there are information about angle and magnitudes of the vector.

Method 2: Triangle of vectors and polygon of vectorIts the drawing method. The drawing of the vectors need to be draw according scale and angle given in the question. The instrument used for this drawing are:(d) ruler(e) protractor(f) sharp pencilUse a suitable scale to draw the first vector.From the end of first vector, draw a line to represent the second vector. (attaching the head with the its tail)Complete the triangle/polygon. The line from the beginning of the first vector to the end of the second vector represents the sum in magnitude and direction.

Example 1.7Example 1.8

Method 3 : Component MethodIts is a calculation method , because every vector can bereplace intox-component and y-component. Replacing a single vector into its components is called Resolving.To determine the resultant of the vector using this method, its need to follow these four keyword carefully.1. Axis2. Resolve vector3. add vector component4. ResultantAxisNeed to be determine before resolving the vector.Resolving(leraian)vectorThe vector that is not on any axis have to be resolve into its component. Resolving vector mean resolving :(a) magnitude(b) DirectionA vector R can be considered as the two vectors. R refers to the resultant vectors. There are two mutually perpendicular component Rx and Ry

Add Vector ComponentandOnly the same axis component can be added.ResultantMagnitude,and Direction of R,Example 7The figure shows 3 forces F1, F2and F3acting on a point O. Calculate the resultant force and the direction of resultant.

(d) Multiplication of vectorIts have been discuss about subtraction and addition of the vector. From subtraction and addition of vector we can explain most of the physical quantity. Now is about multiplication of vectors. When two vectors were multiply the result is called product.There are two kind of product produced :1. Dot Product2. Cross ProductDot ProductThe dot product is fundamentally a projection.

The dot product of a vector with a unit vector is the projection of that vector in the direction given by the unit vector. This leads to the geometric formula

Furthermore,it follows immediately from the geometric definition that two vectors are orthogonal if and only if their dot product vanishes, that is

Cross ProductThe cross product is fundamentally a directed area.

whose magnitude is defined to be the area of the parallelogram?. The direction of the cross product is given by the right-hand rule, so that in the example shownpoints into the page.In mathematics and physics, theright-hand ruleis a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century.

Unit vectorWhen comes into multiplying vector its easier to used component method. The basis for the coordinate system used in vector notation is unit vector.in mathematics, aunit vectorin a normed vector space is a vector whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this:(pronounced "i-hat"),and. The,andis use in 3D or cartesian coordinate and :andis use in Euclidean space.The operation on the vector will be much more faster compared to the drawing method.A vector can be represent in component method asmeaning that a vector A is stretch from origin to point (2,3) in Euclidean space.

1.4 MetrologyMetrology is the science ofmeasurementand its application.Terminology related to measurement uncertainty is not used consistently among experts. To avoid further confusions lets refer to BIPM-VIM(International Vocabulary of Basic and General Terms in Metrology)and GUM (Guide to the expression of uncertainty in measurement).

1.4.1 ErrorVIM define the error as below:error (of measurement)[VIM 3, 2.16] measured quantity valueminus areference quantity valuethere are two type of error(a) Systematic ErrorCharacteristics of systematic error in the measurement of a particular physical quantity:-Its magnitude is constant.-It causes the measured value to be always greater or always less than the true value.Corrected reading = direct reading systematic ErrorSources of systematic Error: Zero Error of instrument. Incorrectly calibrated scale of instrument. Personal error of observer, for example reaction time of observer. Error due to certain assumption of physical conditions of surrounding for example, g = 9.81 ms-2Systematic error cannot be reduced or eliminated by taking repeated readings using the same method, instrument and by the same observer.

(b) Random ErrorCharacteristics of Random Error : Its magnitude is not constant. It causes the measured value to be sometimes greater and sometimes less than the true value.Corrected reading = direct reading Random ErrorThe main source of random Uncertainty is the observer.The surroundings and the instruments used are also sources of random error.Example of random Error: Parallax Error due to incorrect position of the eye when taking readingParallax Error can be reduced by having the line of sight perpendicular to the scale reading. Error due to the inability to read an instrument beyond some fraction of the smallest divisionReading are recorded to a precision of half the smallest division of the scale.Random Error can be reduced by taking several readings and calculating the mean.Error contributes to but is different from Uncertainty

1.4.2 TheUncertainty of the InstrumentalVIM define the Uncertainty as belowuncertainty of measurement[VIM 3, 2.6]non-negative parameter characterizing the dispersion of thequantity valuesbeing attributed to ameasurand(quantity intend to measure), based on the information used and its have a statistical concept of standard deviation means.Instrumental MeasurementWhen handling the experiment the reading is given by the apparatus used, these apparatus have their own uncertainty.instrumental measurement uncertainty(VIM 3, 4.24) the amount (often stated in the form dx) that along with the measured value, indicates the range in which the desired or true value most likely lies.Instrumental measurement uncertainty is used in aType B evaluation of measurement uncertaintyHere the magnitude ofdxis called theabsolute Uncertainty. Absolute Uncertainty is the smallest scale of the instrument or half of the smallest scale if its can be determine easily.InstrumentsAbsolute UncertaintyExample of readings

Millimeter ruler0.1 cm(50.1 0.1)cm

Vernier caliper0.01 cm(3.23 0.01)cm

Micrometer screw gauge0.01 mm(2.63 0.01)mm

Stopwatch (analogue)0.1 s(1.4 0. 1 )s

Stopwatch(Digital)0.01 s(1.452 0.01)s

Thermometer0.5 C(28.0 0.5)C

Ammeter (0 3A)0.05 A(1.70 0.05)A

Voltmeter (0 5V)0.05 V(0.65 0.05)V

The smaller absolute uncertainty of the instrument is contribute to thehigh accuracy, precision and sensitivityof the measuring system of the experiment.1.4.2Analysing Uncertaintyof the data specifically Uncertainty analysing is refer to Uncertainty that cause byrepetition measurementto produce more accurate data. Meaning that if we want to measure a mass of cube, of course we cannot just used a single measurement then we will get the answer. We have to measure the mass with the triple balance beam more than one time for example 3 time. While doing the measurement actually we have continually increasing the Uncertainty. It is a good idea to mention the Uncertainty for every measurement and calculation. In this subtopic we deal with the repetition reading or data. Its known that if we have more than one reading so the true value is the mean of the reading.Mean value for a is Mean value of Uncertainty of a,should be caculated this way1. Calculated the deviation of every data given:

2. Find the sum of deviation

3. find the mean of deviationIts known that the mean deviataion is equally the same as the Uncertainty of the mean value(true value).Or

Working example on a single quantity :1.Aim :to determine the diameter, d of a wire2.Theory: used outer jaw of vernier caliper3.Precaution: measure more than one reading4.Choosing Apparatus and Determine the absolute uncertainty:InstrumentsUncertainty(Absolute/actual)

Vernier caliper0.01 cm

5.Manage the reading/data:Diameter ,d of a wire was measured several time to reduce the Uncertainty and the reading is given in the table below. Find the true value(mean value) and the Uncertainty of the diameter.iiiiiiivvvi

(d0.01)/cm1.551.521.541.531.541.53

6.Determine the quantity and its uncertaintya. Calculating the true value of diameter (mean value) :

b. Calculating the uncertainty of diameter:

So the diameter of a wire should be written (1.54 0.01)cmNote: calculating the uncertainty this way is refer to a single quantity and not involving with the graph.Primary data and secondary data Primary data are raw data or readings taken in an experiment. Primary data obtained using the same instrument have to be recorded to the same degree of precision i.e to the same number of decimal places. Secondary data are derived from primary data. Secondary data have to be recorded to the correct number of significant figures. The number of significant figures for secondary data may be the same (or one more than) the least number of significant figures in the primary data. Measurement play a crucial role in physics, but can never be perfectly precise.It is important to specify the Uncertainty or Uncertainty of a measurement either by stating it directly using the notation, and / or by keeping only correct number of significant figures.Example:51.20.1Processing significant figures Addition and subtractionWhen two or more measured values are added or subtracted, the final calculated value must have the same number of decimal places as that measured value which has the least number , of decimal places.Example1. a = 1.35 cm + 1.325 cm= 2.675 cm= 2.68 cm2. b = 3.2 cm 0.3545 cm= 2.8465 cm= 2.8 cm3. c == 1.142 cm= 1.14 cmMultiplication and division When two or more measured values are multiplied and/or divided, the final calculated value must have as many significant figures as that measured value which has the least number of significant figures.Example1. Volume of a wooden block = 9.5 cm x 2.36 cm x 0.515 cm= 11.5463 cm3= 12 cm32. If the time for 50 oscillations of a simple pendulum is 43.7 s, then the period of oscillation = 43.7 50 = 0.874 s3. The gradient of a graphNote: Sometimes the final answer may be obtained only after performing several intermediate calculations. In this case, results produced in intermediate calculations need not be rounded off. Round only the final answer.1.4.3 Analysing Uncertainty of combining measurement or equation.1. Actual Value is in the scale reading (pointer reading) of an instrument.(single reading)Or is in the mean value.(of the repetition reading)2. Fractional and percentage Uncertainty,(a) The fractional Uncertainty of R :(b) The percentage Uncertainty of R :3. Consequential Uncertianties/Uncertainty- to state the Uncertainty of a derive quantitiesGivenR1 DR1= Data Absolute Data Uncertainty = 51.2 0.1R2 DR2= Data Absolute Data Uncertainty = 30.1 0.1(a) AdditionW = R1+ R2= 51.2 + 30.3 = 81.3DW = DR1+ DR2= 0.1 + 0.1 = 0.2So W DW = 81.3 0.2(b) SubtractionS = R1 R2= 51.2 30.3 = 21.1DS = DR1+ DR2= 0.1 + 0.1 = 0.2So S DS = 21.1 0.2(c) ProductP = R1 R2= 51.2 30.3 =1541.12From

P DP = 1541.12 7.71(d) Quotient

From

Q DQ = 1.70 0.01Working example:1.Aim :to determine the value of B2.Theory:B is given by

3.Precaution: B have a combine uncertainty from various apparatus (quantity)4.Choosing Apparatus and Determine the absolute uncertainty:

QuantityInstrumentsUncertainty(Absolute/actual)

a,bmeter ruler1 cm

qStopwatch(Digital)0.01 s

5.Manage the reading/data:After the measuring and calculating the uncertainty of the quantity a,b,d,q and T(refer 1.4.2). The true value (mean value) and the uncertainty of the quantities are witten as below :a =(1.830.01)m,b=(1.65 0.01) m,d=(0.001060.00003)m,q = (4.28 0.05) sT = (3.7 0.1) x 103s.T is6.Determine the quantity and its uncertainty(a) Find B use the equation given

B = 7.8 x 10-11m3s(b) Find the uncertainty of B1. Fisrt check the equation for addition and subtraction, by applying 1.4.3 no 3 (b) , subtraction so (a b) = (0.180.02)m2. Second calculate the percentage uncertainties in each of the 4 terms:TermMagnitude and uncertaintyFractional UncertaintyUncertainty percentage

(a b)= (0.180.02)m11%

d= (0.001 06 0.000 03) m3%

q= (4.28 0.05) s1.2%

T= (3.70.1) x 103s3%

The Uncertainty in (a b) is now very large, although the readings themselves have been taken carefully. This is always the effect when subtracting two nearly equal numbers. The percentage Uncertainty in d2will be twice the percentage Uncertainty in d; The percentage Uncertainty inwill be half the percentage Uncertainty in T because a square root is a power of.This gives:Uncertainty percentage in B = 11% + 2(3%) + 1.2% +(3%) = 19.7% 20%This gives B = (7.8 1.6) x 10-11m3s-1.the rules for uncertainties therefore :OperatorUncertainty

addition and subtractionADD absolute uncertainties

multiplication and divisionADD percentage uncertainties

powersMultiply the percentage Uncertainty by the power

Example 8The diameter of a cone is (98 1)mm and the height is (224 1 )mm. What is:(a) The absolute Uncertainty of the diameter.(b) The percentage Uncertainty of the diameter.(c) state the volume of the cone and its uncertainty. Give your answer to the correct number of significant number.Example 9Discuss the ways of minimizing systematic and random ErrorExample 10The period of a spring is determined by measuring the time for 10 oscillations using a stopwatch. State a source of:(a) Systematic Error(b) Random Error1.4.4.Method to find Uncertainty/Uncertainty from a graph

Figure 1where n is the number of points plotted.1. The usual quantities that are deduced from a straight line graph are(a) the gradient of the graph m, and the intercept on the y-axis or the x-axis(b) the intercepts on the axes.First calculate the coordinates of the centroid using the formulawhere n is the number of sets of readings5,6.2. The straight line graph that is drawn must pass through the centroid Figure . The best line is the straight line which has the plotted points closest to it. This line will givethe best gradient together with c.3. Two other straight lines, one with the maximum gradientand another with the least gradient, are then drawn. For a straight line graph where the intercept is not the origin , the three lines drawn must all pass through the centroid. Here also we can findand4. To find the Uncertainty for the gradient and intercept used this equationandWorking Example1.AimTo determine the acceleration due to gravity using a simple pendulum.2.Theory: the theory of the simple pendulum, the period T is related to the length l, and the acceleration due to gravity g by the equation

Hence, the acceleration due to gravity,A straight line graph would be obtained if a graph ofagainstis plotted.3.Precaution:The time t for 50 oscillations of the pendulum is measured for different lengths l of the pendulum. The period T is calculated using

4.Choosing Apparatus and Determine the absolute uncertainty:InstrumentsUncertainty(Absolute/actual)

Millimeter ruler0.1 cm

Stopwatch (analogue)0.1 s

5.Manage the tableNote the various important characteristics when tabulating the data as shown in Table

Table 1(a) Name or symbol of each quantity and its unit are stated in the heading of each column. Example: Length and cm, and T(s). The Uncertainty for the primary data, such as length and t time for 50 oscillations, is also written. Example: (l 0.05) cm and (t 0.1)s.(b) All primary data, such as length and time, should be recorded to reflect the precision (absolute uncetainty) of the instrument used.For example, the length of the pendulum l is measured using a metre rule. hence it should be recorded to two decimal places of a cm, that is 10.00 cm, and not 10 cm or 10.0 cm.The time for 50 oscillations t is recorded to 0.1 s, that is 32.0 s and not 32 s.The average value of t is also calculated to 0.1 s. The average value of 31.9 s and 32.0 s is recorded as 32.0 s and not 31.95 s.(c) The secondary data such as T and T2, are calculated from the primary data. Secondary data should be calculated to the same number of significant figures as I hat in the least accurate measurement. For example, T and T2, are calculated to three significant figures, the same number of significant figures as the readings of t.(d) For a straight line graph, there should be at least six point plotted. If the graph is a curve, then more points should be plotted, especially near the maximum and minimum points.Note that the graph is plotted with the assumption that the origin (0, 0) is a point.The x-coordinate of the centroid === 50 cmThe y-coordinate of the centroid === 2.00s2The coordinate for the centroid is (50cm, 2.00s2)

Graph 1from the equation

Hence a graph of T2against l is a straight line, passing through the origin, and gradient,

From the graph,gradient of best line,Maximum gradient,Minimum gradient,Absolute Uncertainty in the gradient,

Fractional Uncertainty in the gradient

percentage Uncertainty in gradient

Acceleration due to gravity,Hence the percentage Uncertainty in g is the sum of the percentage Uncertainty in m only because 4p2is a constant.Therefore percentage Uncertainty in gravity,Dg =SUncertainty percentage = 1.88%according to above equationHence acceleration due to gravity,Written in percentage Uncertaintyg = (9.8701.88%) m s2also can be write in absolute Uncertainty

g = (9.9 0.2) m s2Since there is Uncertainty in the second significant figure, the value of g is given to two significant figures.Activities:Experiment number 1 *To determine the density of a substanceRujukan:1BIPM,Www.bipm.org(2011).2Wikipedia contributors, Wikipedia, the Free Encyclopedia (2012).3Wikipedia contributors, Wikipedia, the Free Encyclopedia (2012).4R. Hutching,Physics(Macmillan Education Ltd, Hong Kong, 1990).5H. Ahmad, R.H. Raja Mustapaha, and D. Bradley,Panduan Kaedah Ujikaji(Dewan Bahasa dan Pustaka, kuala lumpur, 1986).6S. Zainal Abidin,Fizik Amali(Dewan Bahasa dan Pustaka, kuala lumpur, 1992).