Question

A horizontal force $F$ acts on a solid sphere of mass $M$ at its centre. Coefficient of friction between the ground and sphere is $\mu$. Let the maximum force $F$ for which there is no slipping be $\frac{n}{2}\mu Mg$. Find $n$.

• A. $5$
• B. $7$
• C. $6$
• D. $3$

Answer 1

The correct option is B $7$

For linear motion of sphere,
$F-f=Ma$.......(1)
For pure rolling motion:
$a=R\alpha \Rightarrow \alpha =\frac{a}{R}$
For rotational motion about the center of the sphere:
$\tau =I\alpha$
$\Rightarrow fR=I\alpha$
$\Rightarrow fR=\frac{2}{5}M{R}^2\times \frac{a}{R}$
($\because I=\frac{2}{5}M{R}^2$ for sphere)
$\Rightarrow f=\frac{2}{5}Ma$
$\Rightarrow Ma=\frac{5}{2}f$......(2)
Putting equation (2) in equation (1)
$F-f=\frac{5}{2}f$
$\Rightarrow f=\frac{2}{7}F$
For no slipping, $f\le \mu Mg$
$\frac{2F}{7}\le \mu Mg$
$F\le \frac{7}{2}\mu Mg$
Thus, $n=7$.