CLASS 11 UNITS AND MEASUREMENTS
UNITS AND MEASUREMENT
These are the power which has to be raised on M (Mass), L (length), &T (time) to represent a physical Quantity.
Ex: (i) Velocity = distance / time
dim (Velocity) = L/T
= [M0 L T-1]
(ii) Acceleration = Velocity / time
dim (acc) = M0 L T-1 / T
= M0 L T-2
Uses of dimensions Equation:
(1) to check the correctness of formula
(2) to derive formula
(3) to convert a quantity from one unit to another
(1) To Prove:-
A formula is correct if dim(LHS) = dim of RHS
Ex: V = ut + at
dim(LHS) = dim(v)
= M0 L T-1
dim(RHS) = dim(v) + dim(a) + dim(t)
= M0 L T-1 + M0 L T-2 + M0 L T0
= M0 L T-1 + M0 L T-1
= M0 L T-1
These are Units so they can't be added same as
= 2kg + 4kg
= 6 kg
(2) DERIVATION OF FORMULA:-
Ques> Derive the formula of time period of pendulum. It depends upon length & Acceleration due to gravity.
Sol. > let T ∝ la gb
M0 L0 T1 ∝ [L]a [L T-2]b
M0 L0 T1 = [M0 La T-2b]
By Principle of Homogeneity
Principle of Homogeneity
2x=24
x = 4
(3) TO CONVERT ONE PHYSICAL QTY INTO ANOTHER PHYSICAL QTY :-
It is based on the fact that magnitude of the physical Quantity remains same.
n₂ U₂ = n₁ U₁
n₂ = n₁ (U₁/U₂)
n₂ = n₁ (M₁/M₂)a (L₁/L₂)b (T₁/T₂)c
Ex) Convert 1 N(MKS) into Dyne (CGS)
n₂ = 1 (1 Kg/1 gm)a (1 m/1 cm)b (1 sec/1 sec)c
n₂ = 1 (1000 gm/1 gm)a (100 cm/1 cm)b (1 sec/1 sec)c
n₂ = 105 gm cm sec-2
= 105 dyne
✤ Different types of Variables & Constants :-
(1) Dimensional Variables: The physical quantities which have no dimensions but have variables values are called dimensional variables
Ex: Area, Volume, Velocity, force.
(2) Dimensionless Variables : The Physical quantities which have no dimensions have variable Values are called dimensionless variables.
Ex: Angle, specific gravity, strain
(3) Dimensional Constant: The physical quantities which possess dimensions & have constant Values are called Dimensional constants.
Ex: Gravitational Constant.
(4) Dimensionless Constants: The constant quantities having no dimensions are called Dimensionless Constant.
Examples: π, e.
❈ LIMITATION OF DIMENSIONAL ANALYSIS:-
1) The method doesn't give any information about Dimensionless Constant k
2) It fails when a physical quantity depends on more than three
Physical quantities.
3) It fails when a physical quantity (e.g., s=ut+1/2 at2) is the sum or difference of two or more quantities
4) It fails to derive relationships which involve trigonometric logarithms & exponential functions.
S.No. Physical Quantity Relation with the Dimensional
Quantities Formula
1. Area l X b L2
2. Volume l X b X h L3
3. Density Mass /Volume ML-3
4. Speed Distance/time LT-1
5. Acceleration Change in velocity/time = / LT-2
6. Momentum mass X velocity MLT-1
7. Force mass X acceleration MLT-2
8. Work Force X distance ML2T-2
9. Energy Amt. of work ML2T-2
10. Power Work / time ML2T-3
11. Pressure Force/Area ML-1T-2
12. Torque Force X Distance ML2T-2
13. Gravitation constant Force X (Distance)2 ML-3T-2 Mass X Mass
14. Impulse Force X time MLT-1
15. Stress Force / Area ML-1T-2
16. Coefficient of elasticity Stress/strain ML-1T-2
17. Surface Tension Force / length MT-2
18. Surface Energy Work/Area MT-2
19. Strain Chain in Dimension Dimensionless Original Dimension
20. Coefficient of Viscosity Force X Distance ML-1T-1 / Original Dimension
21. Angle Arc/Radius Dimensionless
22. Moment of Inertia Mass X(Distance)2 ML2
23. Planck’s constant Energy/Frequency ML2T-1
24. Force Constant Displacement / time MT-2
25. Universal Gas Constant PV/nT ML-1T-2L3
❄ Light year:-
It is the distance Cover by light in one year.
1 ly = distance one year
= 3x108 x 365 x 24 x 60 x 60
= 9.46 x 1015 m
❄ Astronomical Unit:-
It is the main distance b/w centre of earth & Centre of sun.
1 AU = 1.49x1011 m
= 1.5x1011 m
❄ Parsec:-
It is the radius of a circle in which an arc of 1 AU subtend an angle of 1" at its Centre
❄ Fermi:
It is unit to measure the size of nucleus
1fm = 10⁻¹⁶m
❄ Nanometer
1nm = 10⁻⁹m
❄ Micron :
It is use to measure smaller value.
1um = 10⁻⁶m
❄ Pico : 1um=1p
=10⁻¹²m
❄ SIGNIFICANT FIGURES:
Those, are the figure which are used to measure a quantity accurately.
More significate figure, More accuracy
*RULES:-
① All non-zeroes digits are significant.
Example: 2468
Signo. 4
② All zeroes between two nonzero also consider as significant
Example: 2004806
Signo. 7
③ All zeroes in a decimal no. less than one are not significant which is before, the non zero digit.
Example: 0.0016
Signo. 2
(ii) 0.00460
Signo 3
④ (i) 2500
Signo. 2
(ii) 25.00
Signo. 4
(iii) 2.500
Signo. 4
For addition & Subtraction the answer should be in minimum decimal form
Example:
2.11 + 3.062 = 5.172
For Multiplication & Division the answer should be in minimum significant number
Example:
2.4 * 1.4 = 3.2
✤ ROUND OFF:-
① If skip digits is more than 5 then preceding digit is increased by
Example: 2.567
=> 2.57
② If skip digits is lesser than 5, No change
Example: 2.563
=> 2.56
③ If skip digit is 5
① If preceding digit is even, No Change
② If preceding digit is odd, I increase
Example: 2.565
=> 2.56
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