Rotational Motion Question 98
Question: A solid sphere of mass $(M)$ and radius $(R)$ is rotating about its axis with a constant angular velocity $(\omega)$. If the radius of the sphere is halved, what will be the new angular velocity?
Options:
A) $\omega$
B) $\frac{\omega}{2}$
C) $\frac{\omega}{4}$
D) $2\omega$
Show Answer
Answer:
Correct Answer: D
Solution:
The moment of inertia (I) of a solid sphere about its axis is given by $(\frac{2}{5}MR^2)$.
According to the conservation of angular momentum $(L = I\omega = \text{constant})$, if the radius is halved, the new moment of inertia $(I’)$ becomes $(\frac{2}{5}M$ $(\frac{R}{2})^2 = \frac{1}{5}MR^2)$.
Therefore, the new angular velocity $(\omega’)$ must be such that $(I’\omega’ = I\omega)$.
Therefore, $(\frac{1}{5}MR^2\omega’ $=$\frac{2}{5}MR^2\omega)$
Solving for $(\omega’)$, we get $(\omega’ = 2\omega)$.
