Question

Establish the relation between angular momentum (L), moment of inertia (I) and angular velocity (ω) and give the definition of I on the basis of this relation.

Answer 1

Suppose a rigid body is rotating about an axis YY' with angular velocity ω.

Moment of linear momentum about the axis of rotation is called the angular momentum and it is denoted by L.

= p x r

= mv x r

= mrωr

or \(L = mr^2\omega\) 

A body can be supposed to be made of n small particles of masses m₁, m₂, m₃, .........., mn and distant r₁, r₂, r₃, ......., r_n from the axis of rotation. Then

L = L₁ + L₂ + .......... + L_n

= m₁r₁²ω + µ₂r₂²ω + .......... + m_nr_n²ω

= (m₁r₁² + m₂r₂² + .......... + m_nr_n²) ω

\(=\sum_{i = 1}^nm_ir_i^{}2\ \omega\)

or \(L = I\omega\)

i.e., Angular momentum = Moment of inertia x Angular velocity.

If ω = 1 rads^-1, then L = I

i.e., "moment of inertia of a body about a given axis of rotation is equivalent to its angular momentum which is in the equation of unit angular velocity."