Question

The total angular momentum of a body is equal to angular momentum of its center of mass if the body has
A) Only rotational motion
B) Only translational motion
C) Both rotational and translational motion
D) No motion at all

Answer 1

Hint: Angular momentum for every particle is added to get the angular momentum of the body passing through the axis of rotation. A translational motion can be considered as moving in a circle with infinite radius. Motion in a straight line will have linear momentum which can add up to give additional angular momentum.

Complete step-by step solution
Angular momentum of a body is defined as the cross product of linear momentum and its radius, along which the body is moving. It can also be expressed in the terms of particles as the particle having a linear momentum p and its distance from the axis of rotation be r, Then:
 \[L = r \times p\]
The direction of angular momentum is also given by the right-hand thumb rule, i.e. it is perpendicular to both r and p.
Angular momentum of each particle is then added up and the resultant angular momentum is the angular momentum of the whole body, passing through the axis of rotation. If a body has translation motion as well, it will have an additional linear momentum which will result in an additional angular momentum being added up. Therefore, the total angular moment will only be equal to the angular momentum of its center of mass when it is in pure rotational motion.

So, option A is correct.

Note: It should be noted that when there is no motion, the velocity is 0 and hence linear and angular momentum will also be 0.