Question

An ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moving along the diameter, the angular velocity of the disc will be?
A). Remain constant
B). First decreases and then increases
C). First increases and then decreases
D). Increase continuously

Answer 1

Hint: According to law of conservation of angular momentum when the moment of inertia of the rotating system increases its angular velocity decreases. in the same way when the moment of inertia of the rotating system decreases its angular velocity increases.

Formula used:
$L=I\omega =\text{constant}$
$I\propto \dfrac{1}{\omega }$

Complete step by step answer:
In this question we are considering both disc and ant as a system.
The moment of inertia of a rotating disc about an axis passing through its center and perpendicular to its plane is
${{I}_{1}}=\dfrac{1}{2}M{{r}^{2}}\to \left( 1 \right)$
Here$M$ is mass of disc and $r$ is radius of the disc it remains constant
The moment of inertia of inertia of the ant at initial position is
${{I}_{2}}=m{{r}_{1}}^{2}\to \left( 2 \right)$
Here$m$ is mass of the ant and${{r}_{1}}$ is the distance between the ant and axis of rotation it is variable
Initially${{r}_{1}}$ is equal to$r$ .
The total moment of inertia of the system is
$I={{I}_{1}}+{{I}_{2}}$
By substituting the values from equations $\left( 1 \right)\text{ and }\left( 2 \right)$ we get
$I=\dfrac{1}{2}M{{r}^{2}}+m{{r}_{1}}^{2}$
The angular momentum of the system is
$L=I\omega $
Here, $\omega $ is the angular speed of the system.
It remains constant because no external force is acting on the system.
When the ant starts moving towards the center of the disc the distance between the ant and the axis of rotation decreases that means ${{r}_{1}}$ decreases, then the moment of inertia of the system $I$ decreases so the angular velocity of the system $\omega $ increases.
When the ant passes the center of the disc the distance between the ant and the axis of rotation increases that means ${{r}_{1}}$ increases, then the moment of inertia of the system $I$ increases so the angular velocity of the system $\omega $ decreases.
So, in this case, we can conclude that the angular velocity of the disc will first increase and then decrease.

So, the correct answer is “Option C”.

Note:
When no external force is acting on the system, the linear momentum and the angular momentum of the system remain constant. When we are solving the problems based on the law of conservation of angular momentum, we have to consider the whole as a system and choose the position and orientation of the axis of rotation, otherwise, we can't get the answer correctly and, at the same time, we have to observe the terms which are variable and which are constant.