# Relation between Angular Momentum and Moment of inertia

When a body moves along a straight line, then the product of the, mass m and the linear velocity of the body is called the “linear momentum” p of the body (p = m*v), if a body is rotating about an axis, then the sum of the moments of the linear momenta of all the particles about the given axis is called the “angular momentum” of the body about that axis. It is represented by ‘J’.

Let a body be rotating about an axis with an angular velocity ω. All the particles of the body will have the same angular velocity, but their linear velocities will be different. Let a particle be at a distance r1 from the axis of rotation. The linear velocity of this particle is given by

v1 = r1ωIf the mass of the particle be m1, then its linear momentum = m1v1.

The moment of this momentum about the axis of rotational

= momentum * distance

= m1v1 * r1

= m1 (r1ω) * r1

= m1 r21 ω

Similarly, if the masses of other particles be m2, m3, … and their respective distances from the axis of rotation be r2, r3, … the moments of their linear momenta about the axis of rotation will be m2 r22 ω m3 r23 ω, … respectively. The sum of the moments of linear momenta of all particles, that is, the angular momentum of the body is given by

J = m1 r21 ω + m2 r22 ω + m3 r23 ω + …

= (m1 r21 ω + m2 r22 ω + m3 r23 ω + …) ω

= Smr2) ω

But Smr2 is the moment of interia I of the body about the axis of rotation. Hence the angular momentum of the body about the axis of rotation is

J = I * ω

The S.I. unit of angular momentum is Kg m2s-1 of Js. Its dimensional formula is [ML2T-1].

It is clear from the above formula that just as the linear momentum (mv) of a body is equal to the product of the mass of the body and its linear velocity; in the same way the angular momentum ( of a body about an axis is equal to the product of the moment of interia of the body and its angular velocity about that axis.