Units and measurement chapter 2 Class 11th

Chapter–2: Units and Measurements

Need for measurement: Units of measurement; systems of units; SI units, fundamental and derived units. significant figures. Dimensions of physical quantities, dimensional analysis and its applications.

S.No.	Fundamental Quantity	Unit	Symbol
1.	Length	Meter	m
2.	Mass	Kilogarm	kg
3.	Time	Second	s
4.	Temperature	Kelvin	K
5.	Luminous Intensity	Candela	cd
6.	Current	Ampere	A
7.	Quantity of Matter	Mole	mol

S.No.	Supplementary Quantity	Unit	Symbol
1.	Plane angle	Radian	∆
2.	Solid angle	Steradian	∆Ω

Some common practical units

Fermi- small measuring unit for nuclear size ,    1fermi =1 fm =10-15 m

Angstrom = 1 A0 =10-10 m Nanometer = 1nm = 10-9m Micron = 1µm = 10-6m

Light year – Distance travelled by light in vaccum in one year = 1ly = 3.26 x 1015 m

Abbreviation of power 10-

Prefix	value	Prefix	value
Deci (d)	10-1	Deca (da)	101
Centi (c)	10-2	Hecto (h)	102
Milli (m)	10-3	Kilo (k)	103
Micro (µ)	10-6	Mega (M)	106
Nano (n)	10-9	Giga (G)	109
Pico (p)	10-12	Tera (T)	1012
Femto (f)	10-15	Peta (P)	1015
Atto (a)	10-18	Exa (E)	1018

Dimensions/Dimensional Formulae/Dimensional equation

The powers on basic /fundamental units viz mass (M), length (L), time (T) are called dimensions of a physical quantity.

Ex: - Force = mass x acceleration= [M1] x [L1T-2]

S.N.	Physical Quantity	General Formula	Dimensional Formula	Unit
1.	Distance/Displacement	Distance Covered	[ M⁰ L¹ T⁰ ]	m
2.	Speed/Velocity	Distance/time	[ M⁰ L¹ T⁻¹ ]	m/s
3.	Acceleration	a=velocity/time	[ M⁰ L¹ T⁻² ]	m/s²
4.	Volume	V=l x b x h	[ M⁰ L³ T⁰ ]	m³
5.	Area	Area= l x b	[ M ⁰L² T ⁰ ]	m²
6.	Density	Mass/volume	[ M¹ L⁻³ T ⁰]	Kg/m³
7.	Force	F=mass x a	[ M¹ L ¹T⁻² ]	N
8.	Momentum	P=mass x velocity	[ M¹ L ¹T⁻¹ ]	Kg m/s
9.	Gravitational Constant	G =Fr² /m₁ m₂	[ M⁻¹ L³ T ⁻²]	Nm²/kg ²
10.	Plank’s Constant (E=h ν)	Energy / ν	[ M¹ L ²T ⁻¹]	Js
11.	Work	W=F x d	[ M ¹L ²T⁻² ]	J
12.	Power	P=work / Time	[ M¹ L² T⁻3 ]	J
13.	Angle	Θ =arc/radius	[ M⁰ L⁰ T⁰ ]	radian
14.	Wave Length (𝜆)	Length of wave	[ M⁰ L ¹T ⁰]	m
15.	Impulse	Force x time	[ M¹ L¹ T⁻¹ ]	Ns
16.	Pressure(Stress)	Force/area	[ M¹ L⁻¹T⁻²]	Nm⁻²
17.	Acceleration due to Gravity(g)	Change in velocity/ time taken	[ M⁰ L ¹T⁻² ]	ms⁻²
18.	Thrust(Tension)	Force	[ M¹ L¹ T⁻² ]	N
19.	Moment of Inertia (I)	m x r²	[ M¹ L² T ⁰]	kgm²
20.	Moment of Force/torque	Force x distance	[ M¹ L² T⁻² ]	Nm

21.	Angular velocity (w)	Angle / time	[ M⁰ L⁰ T ⁻¹]	Radian/ s
22.	Angular acceleration (α)	Angular velocity/time	[ M⁰ L⁰ T⁻² ]	Radian/ s²
23.	Angular momentum	I x ῳ    (J=mvr)	[ M¹ L² T⁻¹ ]	kgm²/s²
24.	Strain	Change    in dimension/ origional dimension	[ M⁰ L⁰ T⁰ ]	-
25.	Frequency	No of vibration	[ M ⁰L ⁰T⁻¹ ]	Hz
26.	Time Period	Time	[ M⁰ L⁰ T¹ ]	s
27.	Angular Frequency	2π x frequency	[ M⁰ L⁰ T⁻¹ ]	1/radia n
28.	Velocity Gradient	Velocity/distance	[ M⁰ L⁰ T⁻¹ ]	1/s
29.	Coefficient of Viscosity F= η A dv/dx	η = F ×dx/A dv	[ M¹ L⁻¹ T⁻¹ ]	Pascal second
30.	Gas Constant	R= PV /nT	[ M ¹L² T⁻² K⁻¹ Mol⁻¹ ]	J/Kmol
31.	Coeficient of elasticity	Stress/strain	[ M¹ L⁻¹ T⁻2 ]	Nm-2

Example : - Convert 1 Newton into Dyne.

Answer:

Force= [M1L1T-2]    a=1 b=1, c=-2 N2 = N1 × (M1 / M2 )a × (L1 /L2 )b × (T1 /T2 )c

N2 = 1× (1kg /1g)1× (1m/1cm)1 × (1s/1s)-2

= 1 × 103g × 102m × 1

= 105 gm/s

= 105 dyne   

1Newton = 105 dyne

Example: Check the correctness of formula

E= mc2

Answer

Dimension of LHS = [ M ¹L ²T⁻² ] Dimension of RHS = [M ][LT -1]2 = [ M ¹L ²T⁻² ]

since the dimensions on LHS = Dimensions of RHS, Hence the given formula is correct . Example If force F = a+bx2 , find the dimensions of constants a & b.

Answer

[a ]= [F] = [ M ¹L¹T⁻² ] [bx2 ]= [F]

or    [b ] = [F]/ [x2 ]

[b ] = [ M ¹L1 T⁻² ] / [L]2 = [ M ¹L-1 T⁻² ]

Application: To derive relationship among different physical quantities:

Question Derive an expression for the centripetal forces F acting on a particle of mass m, moving with velocity v in a circle of radius r.

Answer:

F α ma.vb.rc equation 1

[ M ¹L1 T⁻² ] = [M]a [LT-1]b [L]c

=Ma. Lb+c. T-b

Comparing both sides , we get 1=a , 1= b+c , -2 = -b

From these , a=1 , b= 2 , c= -1 putting these values in equation 1, we get

F α m1.v2.r-1

F = mv2/r (k=1)

Limitation/ drawback of dimensional analysis

It is unable to find constants in a formula.

It is unable to derive formulae having more terms viz v =u+at, S=ut+at2/2

It is useless on trigonometric functions, Exponential function, Logarithmic function etc.

Classification of physical quantities

1. Dimensional variable- area, volume, velocity, force , coefficient of viscosity

2. Non dimensional variable- angle, specific gravity, strain

3. Dimensional constant – speed of light , gravitational constant, plank constant, permitivity, permeability, boltzmann's constant.

4. Non dimensional constant – pure no like 1,2,3….and e, π etc

Questions:

18. The velocity v of a particle is given in terms of time t by the equation v = at + bt+c. Find dimensions of a, b and c. ans LT-2, L, T

17. Write the dimensional formula for a/b in the relation P = (a-t2 )/bx, where P is pressure, x is the distance and t is time. Ans MT2
16. If force F, acceleration A and time T are basic physical quantities, find dimensions of energy. Ans FAT2
15. A gas bubble from an explosion under water oscillates with a period proportional to Pa db Ec where P is pressure, d is density of water and E is the energy of explosion. Using dimensional analysis find values of constants a, b and c.    ans – 5/6, ½, 1/3                                               
14. If P represents radiation pressure, C represents speed of light and Q energy striking a unit area per second, what are those non zero integers x, y  and z for which PxQyCz is dimensionless? Ans 1, - 1, 1
13. In the real gas equation (P +a/v2)(V – b) = cT, P is pressure, V is volume, T is temperature and a, b, c are constants. What is the dimensions of a? ans ML5T-2
12. If area A, density ρ and velocity v are taken to be the fundamental units, what would be the dimensional formula for force? Ans Aρv2

11. Write dimensional formula for density, acceleration, force, angular speed, work, pressure, momentum, impulse, nuclear energy, angle, refractive index, universal constant of gravitation, electrical resistance, electric potential, frequency and wavelength.

10. I) Find the value of 1 joule in ergs of energy.

II) Convert one dyne into Newton.

9. The centripetal force F on a particle may depend on mass m speed v and radius of the circle r. Find an expression for the force using the method of dimensional analysis.

8. The speed of sound in air v may depend on mass of the air m, wavelength lambda and frequency f. Using the method of dimensions, find an expression for speed of sound.

7. The time period T of simple pendulum may depend upon mass m of its Bob, length of the pendulum L and acceleration due to gravity g. Find a relation for time period using dimensional analysis.

6. Check the dimensional consistency of the following relation in which m is mass, v is speed, g is acceleration due to gravity and h is height

m.v-2 = m.g.h

5. Write the principle of homogeneity of the dimensions of a physical equation.

4. What is meant by the 'dimensions' of a physical quantity. Explain with the help of an example.

3. Write the number of significant figures in the following examples

1. 203
2. 2300
3. 230 m
4. 2.004
5. 2.040
6. 0.004020 cm
7. 3.020 x10-24

2. Write all the SI units of measurement along with their symbols.

1. What are the basic and derived units? Give one example for each.