Question

A solid sphere of radius $R$ is resting on a smooth horizontal surface. A constant force $F$ is applied at a height $h$ from the bottom. Choose the correct alternative

• A. Sphere will always slide whatever be the value of $h$
• B. Sphere will roll without slipping when $h\ge 1.4R$
• C. Sphere will roll without slipping if $h=1.4R$
• D. None of the above

Answer 1

The correct option is C Sphere will roll without slipping if $h=1.4R$

Here by Newton's equation we get $F=ma...\left(i\right)$
We know that, $\tau =I\alpha$
Also for the sphere to roll without sliding, torque about the point $O$
${\tau }_o=Fh=(\frac{2}{5}m{R}^2+m{R}^2)\alpha$
$\Rightarrow Fh=(\frac{7}{5}m{R}^2\times \frac{a}{R})$
As for the condition of pure rolling ,$a=R\alpha$

$Fh=\frac{7}{5}mRa$, Putting the value of $a$ from equation $\left(i\right)$ we get,

$Fh=\frac{7}{5}FR$

$\therefore h=1.4R$

Hence option C is the correct answer.