A solid sphere of mass 3 kg is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is 2/7. What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface? (in N)

$\mathrm{F}+\mathrm{f}=\mathrm{ma}$   ……….(i)

Torque equation about the centre of mass of the sphere 

   F r-f r = $I_{com}\times \alpha$   (Assuming force of friction acts in the forward direction at the point of contact with the surface )

$$\begin{gathered} \Rightarrow F-f = \frac{1}{r}\left[\frac{2}{5}m{r}^2\right]\times \alpha \\ \Rightarrow F-f = \frac{2}{5}m\left(r\alpha \right)..........\left(ii\right) \\ Also a = r\alpha ......\left(iii\right) \\ Solving f = \frac{3}{7}F \\ The sphere will not slip if f\le \mu mg \\ \Rightarrow \frac{3}{7}F\le \mu mg \\ \Rightarrow F\le \frac{7}{3}\times \frac{2}{7}\times 3\times 10 N \\ \Rightarrow F\le 20 N \end{gathered}$$