Physics Made Easy By Ghanshyam Kumar

ANGULAR MOMENTUM, L
Definition
Angular momentum is a physical quantity that describes the rotational motion of an object about a fixed point or axis.
It is the rotational analogue of linear momentum.
Angular momentum is denoted by the symbol, L.

Angular Momentum of a Particle is the product of the position vector, r and the linear momentum, p.
Formula (Vector form)
L = r × p

Where:
r = position vector of the particle from the reference point
p = linear momentum = m v

Magnitude of Angular Momentum
L = m v r sin(θ)
If the particle moves in a circular path (theta = 90°):
L = m v r

Angular Momentum of a Rigid Body
For a rigid body rotating about a fixed axis:
L = I ω

Where:
I = moment of inertia
ω = angular velocity

4. Moment of Inertia (Common Cases)
Point mass at distance r:
I = m r²

Solid disc or solid cylinder about its central axis:
I = (1/2) M R²

Solid sphere about its diameter:
I = (2/5) M R²

Thin ring about its central axis:
I = M R²

5. Relation Between Torque and Angular Momentum
Torque is the rate of change of angular momentum:
τ = dL / dt

If torque is zero, angular momentum remains constant.

Principle of Conservation of Angular Momentum
If the net external torque acting on a system is zero, then the total angular momentum of the system remains constant.

Mathematically:
Initial angular momentum = Final angular momentum
or
I1 ω1 = I2 ω2

Units and Dimensions
SI unit:
kg m² s⁻¹
Dimensions:
[M L² T⁻¹]

EXAMPLES
Q 1.
A Particle of mass 2kg moving in circular path of radius 3m with a velocity of 4 m/s. Calculate the angular velocity.

Solution ⭐
Mass = 2 kg
Velocity = 4 m/s
Radius = 3 m
L = m v r
L = 2 × 4 × 3
L = 24 kg m² s⁻¹

Q2.
The moment of inertia of a rigid body moving with angular velocity of 6 rad/s is 5 kg m². Calculate its angular momentum.

Solution ⭐
Moment of inertia I = 5 kg m²
Angular velocity ω = 6 rad/s
L = I ω
L = 5 × 6
L = 30 kg m² s⁻¹

Applications of Angular Momentum
In Daily Life
• A spinning top remains upright
• A figure skater spins faster when arms are pulled in
• Stability of bicycles and motorcycles

In Astronomy
• Planetary motion around the Sun
• Rotation of Earth and stars
• Formation of galaxies

In Engineering
• Flywheels in engines
• Gyroscopes in navigation systems
• Rotating machines
In Sports
• Diving and gymnastics
• Spin of a cricket or football
• Throwing discus or hammer