Question

A sphere is projected up an inclined plane with a velocity $v_0$ and zero angular velocity as shown. The coefficient of friction between the sphere and the plane is $\mu =\tan \theta$. If the total time of rise of the sphere is $\frac{x{v}_0}{g\sin \theta }$, find the value of $2x$.

Answer 1

Frictional force, $f=\mu N=tan\theta \left(mgcos\theta \right)=mgsin\theta$ .......(1)

$ma=mgsin\theta +f$

$\therefore$ $ma=mgsin\theta +mgsin\theta$

$⟹a=2gsin\theta$

Using: $v=v_o-at$ where $v=0$

$\therefore$ $a=\frac{v_o}{2gsin\theta }$

$⟹x=\frac{1}{2}$$⟹2x=1$