Question

If $\overrightarrow \tau \times \overrightarrow L = 0$ for a rigid body, where $\overrightarrow \tau $ = resultant torque & $\overrightarrow L $ = angular momentum about a point and both are non - zero. Then
A). $\overrightarrow L $ = constant
B). $|\overrightarrow {L|} $=constant
C). $|\overrightarrow {L|} $ will increase
D). $|\overrightarrow {L|} $ may increase

Answer 1

Hint: The cross product of two vectors can be zero only if they are parallel to each other, this means that the angle between vectors should be either zero or pi. To have this condition the modulus of the angular momentum vector should be changing.

Complete solution:
Option D, $|\overrightarrow {L|} $ may increase is the correct answer. Here $\overrightarrow \tau $is the resultant torque and $\overrightarrow L $ is the angular momentum. A vector that is parallel to the angular velocity is called angular momentum. The torque multiplied by the time interval over which the torque is applied creates a change in angular momentum equal to the torque multiplied by the time interval over which the torque is applied.
It is given that $\overrightarrow \tau \times \overrightarrow L = 0$, this is only possible if the angle between them is zero or pi.
$ \Rightarrow \theta = 0$ or $\theta = \pi $
When $\theta = 0$, the angular momentum will increase.
When $\theta = \pi $, the angular momentum will decrease.

Option D is the correct answer.

Note:
The rotational equivalent of linear momentum is angular momentum. Torque, like force, is defined as the rate of change of angular momentum. The sum of all internal torques in any system is always 0, that is in other words, the net external torque on any system is always equal to the total torque on the system. As a result, the total torque on a closed system (where there is no net external torque) must be 0, implying that the system's total angular momentum is constant.