Question

A solid sphere of radius \(r\) is revolving about one of its diameters with angular velocity \(\omega\). If it suddenly expands uniformly so that its radius increases to \(n\) times its original value, then its angular velocity becomes

• A. \(\dfrac{\omega}{n^2}\)
• B. \(\dfrac{\omega}{n}\)
• C. \(n\omega\)
• D. \(2n\omega\)
• E. \(n^2\omega\)

Hint:

If no external torque acts, angular momentum is conserved: \[ I\omega=\text{constant} \]

Answer 1

The Correct Option is A

Moment of inertia of a solid sphere: \[ I=\frac{2}{5}Mr^2 \] If radius becomes \(nr\), then: \[ I'=\frac{2}{5}M(nr)^2=n^2 I \] Using conservation of angular momentum: \[ I\omega = I'\omega' \] \[ I\omega = n^2 I \omega' \] \[ \omega'=\frac{\omega}{n^2} \] Hence, \[ \boxed{(A)\ \frac{\omega}{n^2}} \]