Question

A player strikes a football such that it started spinning in the air with a angular velocity $\omega$ as shown in the diagram Another player of a opposite team punchers the football through a sharp niddle in the air such that radius of the football contaract to 75% of its orginal value. What will be the new angular velocity of the football if weight of the football remains same

• A. $\frac{4}{7}\omega$
• B. $\frac{16}{9}\omega$
• C. $\frac{4}{25}\omega$
• D. $\frac{25}{4}\omega$

Answer 1

The correct option is B $\frac{16}{9}\omega$

$\begin{array}{c}\text{Initial angular momentum of football is}\\ L_i=I_{\omega }=\frac{2}{3}M{R}^2\omega \\ \text{Whose}M\text{is mass of football,}R\text{is initial}\\ \text{radius of football}\\ R^{\prime }=R-\frac{R}{4}=\frac{3R}{4}\end{array}$

$\begin{array}{c}\text{The niddle can be assumed to provide}\\ \text{radial force ie no torque. Thorefore}\\ \text{angulal momentum remains constant}\\ L_i=L_f\end{array}$

$\begin{array}{cc}\frac{2}{3}M{R}^2\omega & =\frac{2}{3}{M}^{\prime }(-\frac{3R^2}{4}){\omega }^{\prime }(M^{\prime }=M)\\ {\omega }^{\prime }& =\frac{16}{9}\omega \\ \text{Hence option}B\text{is correct}& \end{array}$