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  4. A solid sphere of mass M and radius r slips on a rough horizontal plane. At some instant it has translational velocity v0 and rotational velocity about the centre v0 / 2 r. The translational velocity after the sphere starts pure rolling is (x v0/y) in forward direction. Find (x+y).

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Q. A solid sphere of mass $M$ and radius $r$ slips on a rough horizontal plane. At some instant it has translational velocity $v_{0}$ and rotational velocity about the centre $v_{0} / 2 r$. The translational velocity after the sphere starts pure rolling is $\frac{x v_{0}}{y}$ in forward direction. Find $(x+y)$.Physics Question Image

System of Particles and Rotational Motion

Solution:

image
Angular momentum can be conserved about lowest point
$m \,v_{0} r+\frac{2}{5} \,m r^{2} \frac{v_{0}}{2 r}$
$=m\, v' r+\frac{2}{5} \,m r^{2} \frac{v'}{r}$
$\frac{10\, m \,v_{0} r+2 \,M\, v_{0} r}{10}$
$=\frac{7 \,m\, v' r}{5} v'=\frac{6 \,v_{0}}{7}$