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# A solid sphere of mass m is lying at rest on a rough horizontal surface. The coefficient of friction between the ground and sphere is $\mu$. The maximum value of F, so that the sphere will not slip is equal to

$\begin {array} {1 1} (a)\;\frac{7}{5} \mu mg & \quad (b)\;\frac{4}{7} \mu mg \\ (c)\;\frac{5}{7} \mu mg & \quad (d)\;\frac{7}{2} \mu mg \end {array}$

$Ff= Macm$--------(1)
$f_e= I \alpha$
$f= \large\frac{2}{5}$$MR^2.\large \frac{\alpha}{R} f= \large\frac{2}{5} Ma _{cm}------(2)[ a_{cm}=Rd for rolling] \mu mg \geq \large\frac{2}{5}$$m a_{cm}$
$a_{cm} \leq \large\frac{5}{2}$$\mu g From (1) and (2) F= M \bigg[ 1+ \large\frac{2}{5} \bigg]a_{cm} \quad= \large\frac{7}{5}$$ M a_{cm}$
$F_{max }= \large\frac{7}{5}$$M \large\frac{5}{2}$$\mu g$
$\qquad= \large\frac{7}{2}$$M \mu g$