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Units And Dimensions
EDUACHIEVERS ACADEMY
UNITS AND DIMENSIONS Introduction — In physical sciences we deal with large number of quantities which have to be measured. The measurement is basically a comparison with a standard unit. The chosen standard of measurement of quantity which has essentially the same nature as that of the quantity is called unit of the quantity. Although the number of physical quantities is very large, all of them can be expressed in terms of seven basic or fundamental quantities. These seven fundamental quantities, along with their SI units and unit-symbols, are given in Table–1. All other quantities are called derived quantities. In mechanics all the quantities can be expressed in terms of mass (M), length (L) and time (T), in heat we require, in addition, temperature ( or K) and in electricity and magnetism, current (I or A). It should be noted that these quantities are selected for convenience and not through any necessity. Thus in mechanics, we can equally well choose length, force and time as basic quantities, in which case mass would become a derived quantity. Similarly, in electricity and magnetism we can choose charge (Q) instead of current. Table 1 : Quantity Mass Length Time Temperature Electric Current Luminous Intensity Amount of Substance
Table 2 : Abbreviation for Multiples and Sub-multiples Symbol Prefix Multiplier T G M k h da d c m n p
S R
tera giga mega kilo hecto deca deci centi milli micro nano pico
D A C A
Y
1012 109 106 103 102 101 10–1 10–2 10–3 10–6 10–9 10–12
M E
E V
Basic Quantities SI Unit Unit-Symbol kilogram kg metre M second s kelvin K ampere A candela cd mole mol
A U D E
selected which corresponds to the transition 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 of 91926331770.0 periods of oscillation of the selected transition is defined as 1 s.
IE H C
In addition, there are two supplementary units-radian (rad) for plane angle and steradian (sr) for solid angle. The international system of units, abbreviated SI, is an extended version of the MKS (metre, kilogram, second) system. According to this system, the fundamental units of mass, length and time namely kilogram, metre and second have been defined as below : (i) Metre (m) : One metre is defined as the distance containing 1,650,763.73 wavelengths of the orange light emitted by pure krypton-86. (ii) Kilogram (kg) : 1 kilogram is defined as the mass of a particular platinum-iridium cylinder kept at International Bureau of Weights and Measure in Paris. In practice, the mass of 1 litre of water at 4oC is 1 kilogram. On atomic scale, 1 kilogram is also the mass of 5.0188 1025 atoms of 12 6C (isotope of carbon). (iii) Second (s) : Cesium-133 atom emits electromagnetic radiations of several wavelengths. A particular radiation is
Dimensions of a physical quantity — The dimensions of a physical quantity are defined as the power to which the fundamental units of mass, length and time have to be raised to represent a derived unit of the quantity. We use square brackets [ ] to denote "the dimension of" the quantity written parenthesis Ex. Volume = length breadth height [V] = [L] [L] [L] = [L3] Thus, to represent volume, we have to raise [L] to the power 3. Therefore, volume is said to have three dimensions in length. Similarly, velocity =
=
displacement time [ L] [T ]
= [L1 T – 1] = [M 0 L1 T –1]. Hence the dimensions of velocity are : zero in mass, +1 in length and –1 in time. i.e., velocity
Dimensional formula of a physical quantity is an expression which tells us : (i) the fundamental units on which the quantity depends, and. (ii) the nature of the dependence.
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Units And Dimensions
EDUACHIEVERS ACADEMY
Table 3 : Physical Quantities, Relation with Quantities, Dimensional Formula and S.I. Units S.N.
Physical Quantity
Relation with other Quantities
Dimensional Formula 2
S.I. Units
0 2 0
[L] [L] = [L ] = [M L T ] [L] [L] [L] = [L3] = [M0L3T0]
m2 m3
1. 2.
Area Volume
length breadth length breadth height
3.
Density
mass volume
4.
Specific gravity
density of water at 4 oC
5.
Speed or Velocity
distance time
[L] = [LT T –1] = [M0L1T–1] [T]
6.
Linear Momentum
mass velocity
[M] [LT –1 ] = [M1L1T–1]
Acceleration
change in velocity time taken
Acceleration due to gravity (g)
change in velocity time taken
[L T ] = [LT T–2] = [M0L1T–2] [T]
ms–2
Force Impulse
mass acceleration force time
[M] [LT –2] = [M1L1T–2] [MLT –2] [T]= [M1L1T–1]
N (newton) Ns
M 1 3 0 L3 = [M L T ]
density of body
7.
8. 9. 10.
= 1 = [M0L0T 0]
L
S R
[MLT 2 ]
Pressure
force/area
12.
Universal constant
From Newton’s law of gravitation
E V
IE H C F=
13. 14.
Work
A U D E
Gm m
D A C A
No units
M E
T = [LT T –2] = [M0L1T–2] T
11.
of gravitation (G)
kg m –3
Fr or G = m m
r where F is force between masses m1, m2 at a distance r. force displacement
= [M1L–1T –2]
[L2 ]
G
[ MLT 2 ] [ L2 ] [MM]
Y
ms–1 kg ms–1 ms–2
Nm–2
Nm 2 kg–2
= [M–1L3T –2]
[MLT –2] [L] = [M1L2T –2] 1
2
–2
J (joule)
15.
Energy (including Potential Energy, Kinetic Energy, Heat Energy, Light Energy etc.,) Moment of force
16.
Power
work time
[ML2 T 2 ] = [M1L2T –3] [T]
W (watt)
17.
Surface tension
force length
[ML2 T 2 ] = [M1L0T –2] [T]
Nm –1
Surface energy
energy area
[ML2T 2 ]
Force constant
force displacement
18.
19.
[M L T ]
J
force displacement
[MLT –2] [L] = [M1L2T –2]
Nm
E=W
[L2 ]
= [M1L0T –2]
Jm –2
MLT –2 L
= [M1L0T –2]
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Nm –1
2
S.N. 20. 21.
Physical Quantity
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Units And Dimensions
EDUACHIEVERS ACADEMY Relation with other Quantities
Dimensional Formula
S.I. Units
1 1 –2
Thrust Tension
force force
[M L T ] [M1L1T–2]
22.
Stress
Force area
[MLT 2 ]
23.
Strain
change in configuration original configuration
[L] = 1 = [M0L0T0] [L]
No unit
24.
Coefficient of elasticity
[ M1L1T 2 ] = [M1L–1T–2]
Nm–2
25. 26.
Radius of gyration (K) Moment of inertia (I)
stress strain distance Mass (distance)2
[L] = [M0L1T0] [ML2] = [M1L2T0]
m kg m2
27.
Angle ()
length (l) / radius (r)
[L] = 1 = [M0L0T0] [L]
28.
Angular velocity ()
angle () time (t )
1 = [T–1] = [M0L0T–1] [T]
29.
Angular acceleration ()
[ 1T ] = [T–2] = [M0L0T–2] [T]
rad s–2
30. 31. 32.
Angular Momentum Torque Wavelength ()
change in angular velocity time taken I I/T length of one wave, i.e., distance,
[ML2] [T–1] = [M1L2T–1] [ML2] [T–1] = [M1L2T–2] [L] = [M0L1T0]
kg m2 s–1 Nm m
33.
Frequency ( )
number of vibrations/sec
1 = [T–1] = [M0L0T–1] [T]
s–1or Hz (Hertz)
34.
Velocity of light in a medium
[L] = [M0L1T–1] [T]
ms–1
35.
Velocity gradient
velocity distance
[LT 1 ] = [T–1] = [M0L0T–1] [L]
s–1
36.
Rate of flow
volume time
[L3 ] = [L3T–1] = [M0L3T–1] [T]
m3 s–1
37.
Coefficient of viscosity
Force Area velocity gradient
[ML–1 T–1]
kg m–1 s–1
38. 39.
Temperature Heat (Q)
Energy
K = [M0L0T0K1] [M1L2T–2]
K J
40.
Specific heat (s)
s
[ML2 T 2 ] = [M0L2T–2K–1] [MK]
J kg–1 k–1
41.
Latent heat (L)
Q M
[ML2 T 2 ] = [M0L2T–2] [M]
J kg–1
Gas constant (R)
pressure volume temperature
[ML1T 2 ] [ L3 ] = [M2L2T–2K–1] [K]
J K–1
42.
A U D E
[L2 ]
E V
IE H C distance travelled time taken
Q m
S R
N N = [M1L–1T–2]
D A C A
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Nm–2
M E
Y
rad
rad s–1
3
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Units And Dimensions
EDUACHIEVERS ACADEMY S.N.
Physical Quantity
Relation with other Quantities
Dimensional Formula
S.I. Units
43.
Boltzmann constant (A)
pressure volume temp. Avo. No.
[M1L2T –2K–1]
J k–1
44.
Planck’s constant (h)
energy frequency
45.
Charge
I
Q t
[IT] or [Q]
coulomb
46.
Potential
V
W Q
[ML2 T 2 ] = [ML2T–3I–1] [IT]
J/C or volt
Electric field
F E q
Capacitance
q q2 C V W
[ML2 T 2 ]
[q 2 ]
[I 2 T 2 ]
49.
Permittivity
0
50.
Resistance
R
V W I QI
51.
Conductance
C
R
52.
Resistivity
53.
Conductivity
54.
Inductance
47.
48.
A U D E
[ML2 T 2 ] [T 1 ]
[MLT 2 ] = [MLT T–3I–1] [IT]
S R
J-s
M E
= [M–1L–2T4I2]
[MLT 2 ] [L2 ]
= [M–1L–3T4I2]
[ML2 T 2 ] = [ML2T–3I–2] [IT] [I]
E V
IE H C A l
D A C A
[I 2 T 2 ]
[ F ] [r 2 ]
R
= [M1L2T–1]
1
dt W T L e dI Q l
1
[ML T 3 I 2 ] 2
= [M–1L–2T3I2]
Y
N/C or
V m
C/V or farad
C/Vm
V/amp or ohm
(ohm)–1
[ML2 T 3 I 2 ] [L2 ] = [ML3T–3I–2] [L]
ohmmetre
[M–1L–3T3I–2]
(ohmmetre)–1
[ML2T 2 ] [T] = [ML2T–2I–2] [IT] [I]
V/sec/amp. or Henry
ML2 T 2 T = [ML2T–2I–1] [IT]
55.
Magnetic Flux
= e dt
56.
Magnetic Induction
B
A cos
[ML2 T 2 I 1 ]
Magnetic Field Intensity
H=
B 1 Id sin 4 r 2
[ I ] [ L]
57.
[L2 ]
[L2 ]
= [MT–2I–1]
= [IL–1]
volt sec = weber
Wb/m2 or tesla
ampere turns/m or oersted
58.
Permeability
Fr
I l
[MLT 2 ] [L2 ] [I 2 ] [L2 ]
= [MLT T–2I–2]
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henry/metre
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Use of Dimensional Analysis — In a correct physical equation, the dimensions of all the terms must be the same. This is called the principle of homogeneity of dimension. This principle is based on the fact that only similar quantities can be added or subtracted and leads to the following simple applications of dimensional analysis. (1) Conversion of Units — Consider a physical quantity having dimensional formula MaLbTc. Let there be two systems of units (M1, L1, T1) and (M2, L2, T2). If n1 and n2 be the numerical values of the quantity in the two systems respectively, then since they represent the same quantity
n M a Lb Tc n M a Lb Tc M n2 n1 1 M2 The ratios
a
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Units And Dimensions
EDUACHIEVERS ACADEMY
L1 L 2
b
T1 T 2
c
(3) We cannot derive the formulae containing trigono-metrical functions, exponential functions, log functions etc., which have no dimensions. (4) This method cannot be used to derive an exact form of relation, when it consists of more than one part on any side. (5) It gives no information whether a physical quantity is a scalar or vector. Errors in Measurement — Whenever a quantity is measured there is always some uncertainty or error. Even with a faultless instrument there is always some random error due to lack of perfection of the observer and the sensitivity limit (or least count) of the instrument. If a quantity is obtained by measurement of some other quantities, then the errors in all the measured quantities have to be combined. This is done as follows: Suppose a quantity X depends on three quantities A, B and C as
M L T , , are called the conversion factors M L T
for mass, length and time respectively. For conversion from MKS to CGS system M 1 1kg L 1m 103 , 1 102 M2 1g L2 1cm
E V
Thus n2 = n1 103a + 2b (2) Checking the correctness of equations. (3) Deriving simple relations connecting physical quantities provided all the quantities on which a certain quantity depends are known. Limitations of Dimensional Analysis (1) This method gives us no information about dimensionless constants in the formula. (2) If a quantity depends on more than three factors having dimensions, the formula cannot be derived.
A U D E
IE H C
kAm B n Cr
where k is a constant. If A, B, C, X are in errors is A, B, C and X, respectively, then it can be shown that
S R
T1 1s 1 T2 1s
D A C A X
M E
Y
X A B C m n r X A B C
This gives fractional error in X. The maximum possible fractional error is obtained by taking all signs as positive. A B C X m n r A B C X max
Therefore, maximum percentage error in X is B C A n r m A B C
or PX = mpA + npB + rpC where 'P' stands for percentage error.
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