A solid sphere of mass M and radius r shown in figure slides along a rough horizontal plane. At some instant it has translational velocity V_o and rotational velocity about the centre $\frac{V_{o}}{2 r}$ . The translational velocity after the sphere starts pure rolling is $\frac{6 V_{0}}{N}$ . The value of N is ….

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answer is 7.

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Detailed Solution

No external torque; so angular momentum is conserved
L_i=L_f
$\begin{matrix} MVor + IωO = MVr + Iω \end{matrix}$
$\begin{matrix} MVor + \frac{2}{5} {Mr}^{2} \cdot \frac{VO}{2 r} = MVr + \frac{2}{5} {Mr}^{2} \cdot \frac{V}{r} \end{matrix}$ (Pure rolling)
$\begin{array}{l} VO + \frac{VO}{5} = V + \frac{2 V}{5} \\ V = \frac{6 Vo}{7} \end{array}$