CHAPTER 1 MEASUREMENTS & UNITS

1.1:Physical Quantities & Units

• A physical quantity is a term (a word or an expression) that is used to define or to describe a physical property of an object or system.

• Examples:• -Length:define the distance between two points.• -Mass: define the amount of substance in an object.• -Time: define the duration of an incident or occurrence.• -Speed:define as the rate of change of distance in a

certain time interval.• -Stress : define as the force per unit cross-sectional

area.

• Every physical quantity can be measured with a measuring scale and given a unit.

• A unit is the standard used to compare the magnitude or measurement of a physical quantity.

• All physical quantities can be divided into 2 categories:

Base Quantities & Base Units

• Base Quantities are physical quantities that measures the most basic or fundamental properties of the body or system.

• Base quantities are: length,mass,time,current,temperature, amount of substance and light intensity.

• Each of these base quantities are measured using the following base units:

Base Quantity S.I Unit Symbol

Length metre m

Mass kilogram kg

Time second s

Electric current ampere A

Temperature kelvin K

Amount of Substance

mole mol

Light Intensty candela cd

• Each of these standard base units are defined:• Example:1 metre is equal to 1650763.73

wavelengths of light emitted from the krypton-86 atom.

• 1 kilogram is the mass of a platinum-iridium cylinder kept at the International Bureau of Weights and Measures,Sevres,France.

• 1 second is the time taken for 9192631770 vibrations of the light emitted by a caesium-133 atom.

Homework

• What is the definition for the following units:

• (a)Kelvin

• (b)Ampere

• (c)Mole

• (d)Candela

Derived Quantities & Derived Units

• Physical quantities that are related to one or more base quantities are called derived quantities.

• Example:• -area,A is length x length • -speed,v is rate of change of

distance/time.• -volume,V is area x length• -density,ρ is mass per unit volume.

• The derived units for the derived quantities are obtained from the relation of the quantity to the base quantities;

• Example:

• -unit of area,A is m x m =m2

• -unit of speed,v is m/s

• -unit of volume,is m2xm=m3

• -unit of density is kg/m3=kgm-3

Q1.

• What is the SI unit for

(a) acceleration?

(b) force?

1.2:Dimensions of Physical Quantities.

• The dimension of a physical quantity is the relation between the physical quantity and the base quantities.

• The dimension of a physical quantity is stated in terms of M,L,T,A,θ,N and C which are the dimensions for the base quantities

• The symbol for the dimension of a physical quantity is represented by:

[physical quantity]

Dimension of Base Quantities

• Dimension of mass=[mass]=M

• [length]=L

• [time]=T

• [electric current]=A

• [temperature]=θ

• [mole]=N

• [light intensity]=C

Remember

• For numerical constant of proportionality without unit, the [constant]=1

• Example:[π]=1

• To find the dimension of a derived quantity, start from its definition.

Rules of Operation For Dimension

• The operation for dimension follows the rules of operation for multiplication and division.

• Dimension for two different physical quantities cannot be subtracted or added.

• Two physical quantities can be added or subtracted if they are of the same unit or dimension.

Dimension of Derived Quantities.

• Write down the dimension of the following quantities:

• (a) volume

• (b) density

• (c) velocity

• (d) acceleration

• (e) force

• (a) volume = length x length x length

[volume]=[length x length x length]

=[length]x[length]x[length]

=L x L x L

[volume]=L3

• (b)

3

3

][

][

][][

MLdensity

L

M

volume

massdensity

volume

massdensity

• (c)

1][

][

][][

LTvelocity

T

L

time

ntdisplacemevelocity

time

ntdisplacemevelocity

• (d)

2

1

][

][

][][

LTonaccelerati

T

LT

time

velocityonaccelerati

time

velocityofchangeonaccelerati

• (e)

2][

][][][

MLTforce

onacceleratimassforce

onacceleratimassforce

Q2:

• What is the dimension and unit for energy?

Q3:

• What is the dimension for :

• (a)the coefficient of static friction,μ?

• (b) pressure and stress.

Q4:

• Van der Waal’s equation for the pressure of a real gas is given by the relation:

• where P = gas pressure,• V= volume of gas,

R = molar gas constant, T= temperature in Kelvin

a,b = dimensional constants n= number of moles of gas.What is the dimension and unit for a ,b and R?

nRTbVV

aP ))((

2

Answer to Q3:

25

25

2321

22

222

][

)(

]][[]][[][

][

][][][)(

skgmaforunit

TMLa

LTML

VPVPa

V

a

V

aP

V

aP

3

3][][)(

mbforunit

LVbbV

11

1122

1122

321

2

][

]][[

]][[][

][]][[

KJmolor

KmolskgmRforunit

NTMLR

N

LTML

Tn

VPR

nRTbVV

aP

Q5

• The power P required to overcome external resistances when a vehicle is travelling at a speed v is given by the expression , P = av + bv2 where a and b are constants. Derive the dimensions for the constants a and b. Then deduce the units for a and b in terms of the base SI units.

• Ans:[a]=MLT-2 ,[b]=MT-1

1.3:Uses of Dimensions

• (1)To determine the unit for derived quantity.

Example:

PaPascalorNmorskgmpressureforunit

TMLL

MLT

area

forcepressure

NNewtonorkgmsforceforunit

MLTforce

msonacceleratiforunit

LTonaccelerati

,

][

][][

,

][

][

221

212

2

2

2

2

2

Answer:

• [energy]=[F][s]=MLT-2L=ML2T-2

Unit of energy =kgm2s-2

(2)To Check Dimensional Homogeneity Of An Equation.

• For a true or correct equation,the dimensions of all terms in the equation are the same,

• or the dimension on the left side of the equation and the dimension on the right side of the equation are the same.The equation is said to be dimensionally consistent or homogeneous.

Fact 1:

• All correct equations are dimensionally consistent.

• Example :

consistentensionallyisasuv

TLLLTsaasas

TLLTuu

TLLTvv

correctisasuv

dim2

)(][][][]2[

)(][][

)(][][

2

22

222

222122

222122

22

Fact 2:

• A dimensionally consistent equation is not necessarily correct because the value of constant of proportionality can be wrong.

• Example 1:

• but dimensionally consistent.

22

222

22

][

][

][][

4

1

2

1

TLT

L

g

l

Ttt

g

lt

correctnotisg

lt

The correct equation is

g

lt 2

Fact 3:

• A dimensionally consistent equation is not necessarily correct because it can be incomplete or has extra terms.

• Example 2:

• but dimensionally consistent.

.22

222 correctnotis

t

sasuv

222

222 ][]2[][][ TL

t

sasuv

Fact 4:

• An equation that is dimensionally not consistent is not correct.

consistentnotensionallyisasuv

TLLLTsaasas

LTu

LTv

correctnotisasuv

dim2

)(][][][]2[

][

][

2

222

1

1

(3):Derivation of Physical Equation

• An equation relating a physical quantity to other known physical quantities can be derived by the method of dimension.

Example :

• The period of oscillation t of a simple pendulum is dependent on its length,l and the acceleration due to gravity g.

• Assume that the period is given by:

yxglt

alityproportionoftconsensionlesstheiskwhere

gklt yx

tandim

yyxyx

yx

TLLTLT

kwhereglkt22 )(

1][,][][][][

• Equating indices of T and L:

g

lkt

gkltHence

yxyx

y

y

21

21

,

2

1)2

1(0

2

1

21

• The value of k, the constant of proportionality can only be determined by conducting an experiment.

Q6

• The frequency f of the note produced by a stretched string depends on its length L, the tension T of the string and the mass per unit length m of the string. Use the method of dimension to derive an equation for f.

• Ans: m

T

L

kf

Self-Test

• Q7

For an object moving with uniform

Acceleration, the velocity v is given by the

Equation v2=p + qx ,where p and q are

Constants and x is a variable.What is the

dimension of the term qx?

Q8:STPM 2004

• If E is the rotational kinetic energy and L is the angular momentum of a body,the ratio has the same dimension as:

A. Velocity

B. Displacement

C. Frequency

D. momentum

L

E

Q9:STPM 2000

• Which of the following products does not have the same unit as work?

• A. Power x time

• B. Pressure x volume

• C. Torque x angular velocity

• D. Charge x potential difference

• E. Mass x gravitational potential

Q10:GCE A-Level

• The experimental measurement of the heat capacity C of a solid as a function of temperature T is to be fitted to the expression C=αT + βT3. What are the units for α and β respectively.

Q11:GCE A-Level

• The energy of a photon of light of frequency f is given by hf, where h is the Planck constant. What are the base units of h?

Q12:STPM 2006P1Q1:

• Which of the following is not equivalent to the unit of energy?

A. Electron volt (eV)

B. Volt coulomb(VC)

C. Newton metre(Nm)

D. Watt per second(Ws-1)

Q13:STPM 2006 P2Q1:

• (a)Determine the dimension of Young’s Modulus.

• (b)The Young’s Modulus can be determined by propagating a wave of wavelength λ with velocity v into a solid material of density ρ. Using the dimensional analysis, derive a formula for Young’s Modulus

Answers:

• Q7- L2T-2

• Q8-

• Q9-C

• Q10-

• Q11- kgm2s-1 or Js

• Q12- D