Question

A boy sitting firmly over a rotating stool has his arms folded. If he stretches his arms, his angular momentum about the axis of rotation:
A. Increase
B. Decreases
C. Remains unchanged
D. Doubles

Answer 1

Hint: The main concept to apply is the concept of conservation of angular momentum. The angular momentum is analogous to the linear momentum. Angular momentum is defined as the product of moment of inertia and angular velocity.
Angular momentum, $L = I\omega $

Complete step-by-step answer:
The angular momentum is the product of moment of inertia and angular velocity.
$L = I\omega $
So, here we see that the angular momentum varies with the moment of inertia, which represents the distribution of the mass around the central axis and the angular velocity, which is the speed of change of radians subtended at the centre per unit time.
The law of conservation of angular momentum states that unless any external force acts on the rotating body, the total angular momentum remains constant.
Let us take this example of a boy sitting on a rotating stool. He is rotating at a fixed speed.
If he stretches his arms out wide, the moment of inertia increases.
$I = M{R^2}$
We see that the moment of inertia is directly proportional to radius of rotation. Hence, when the boy stretches his hand out, there should be an increase in the moment of inertia and hence, the angular momentum should increase.
But that is incorrect. The angular momentum does not increase.
This is because the angular momentum is constant since there is no external force acting on the boy. So, what exactly happens is that the increase in moment of inertia is compensated by the decrease in angular velocity.
When the boy stretches his hand out, his angular velocity decreases and he turns slower than usual.
Thus, the product of moment of inertia and angular velocity is always constant.
$L = I\omega $
Since L is constant, we can say, $I \propto \dfrac{1}{\omega }$

Hence, the correct option is Option C.

Note:The fact that the angular momentum remains constant under no external influence is used by the ballet dancers to perform a step wherein they rotate about their body as its axis. They are able to increase their speed of rotation if they bring their arms closer to each other, thereby, decreasing their radius of rotation, in turn, the moment of inertia.