Question

A uniform solid sphere of radius $R=25$ cm is projected on a horizontal surface with a velocity $v_0=20$cm/s and simultaneously given a spin ${\omega }_0$. In consequence, the sphere experiences a combined motion. Due to the friction between the sphere and horizontal surface, both translational and rotational motion stop simultaneously. Then, what must be the magnitude of initial spin ${\omega }_0$ (in rad/s)?

Answer 1

Let the time taken for the sphere to stop be $t$.
During this time let it have an acceleration $a$, and an angular acceleration $\alpha$.
The only horizontal force acting is friction $=\mu mg$
The net torque about the centre is $=\mu mgR$
From force equation, $a=\mu g$
From torque equation, $\alpha =\frac{\mu mgR}{I}=\frac{\mu mgR}{\frac{2m{R}^2}{5}}$ = $\frac{5\mu g}{2R}$
We know that time taken, $t=\frac{V_o}{a}=\frac{{\omega }_o}{\alpha }$
${\omega }_o$ = $\left(\frac{\alpha }{a}\right)V_o$

${\omega }_o$ = $\left(\frac{\frac{5\mu g}{2R}}{\mu g}\right)V_o$

${\omega }_o$ = $\left(\frac{5\times 20}{2\times 25}\right)=2$