Transcript
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
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