Question

A solid sphere is rotating about a diameter at an angular velocity $ \omega . $ If it coots so that its radius reduces to $ \frac{1}{n} $ of its original value, its angular velocity becomes

• A. $ \frac{\omega }{n} $
• B. $ \frac{\omega }{{{n}^{2}}} $
• C. $ n\omega $
• D. $ {{n}^{2}}\omega $

Answer 1

Answer: (D)

From law of conservation of angular momentum, if no external torque is acting upon a body rotating about an axis, then the angular momentum of the body remains constant that is
$J=I \omega$
Also, $I=\frac{2}{5} M R^{2}$ for a solid sphere.
Given, $ R_{1}=R, R_{2}=\frac{R}{n} $
$\therefore \frac{2}{5} M R^{2} \omega_{1}=\frac{2}{5} M\left(\frac{R}{n}\right)^{2} \times \omega_{2} $
$\Rightarrow \omega_{2}=n^{2} \omega_{1} $
$=m^{2} \omega$