Question

A sphere of radius $r$ and mass $m$ has a linear velocity $v_{0}$ $m / s$ directed to the left and no angular velocity as it is placed on a horizontal platform moving to the right with a constant velocity $10\, m / s$. If after sliding on the platform the sphere is to have no linear velocity relative to the ground as it starts rolling on the platform without sliding. The coefficient of kinetic friction between the sphere and the platform is $\mu_{k}$. Determine the required value of $v_{0}$ in $m / s$.

Answer 1

$f_{r} \Delta t=I_{ cm } \omega\,\,\,...(i)$
$-f \Delta t=m\left(0-v_{0}\right)\,\,\,...(ii)$
$\omega r=v_{1}\,\,\,...(iii)$
$\Rightarrow m v_{0} r=\frac{2}{5} m r^{2} \omega$
$\Rightarrow \omega=\frac{5 v_{0}}{2 r} $
$\Rightarrow v_{1}=\frac{5 v_{0}}{2}$
$\Rightarrow v_{0}=\frac{2 v_{1}}{5}$
$ \Rightarrow v_{0}=\frac{2 \times 10}{5}=4\, m / s$