Question

A rigid body of mass $m$ and radius $R$ rolls without slipping on a rough surface. A force is acting on the rigid body at distance $x$ from the centre as shown in figure. Find the value of $x$ so that static friction will be zero.

• A. $\frac{I_{\text{cm}}}{2mR}$
• B. $\frac{2I_{\text{cm}}}{mR}$
• C. $\frac{I_{\text{cm}}}{mR}$
• D. $\frac{2I_{\text{cm}}}{3mR}$

Answer 1

The correct option is C $\frac{I_{\text{cm}}}{mR}$

Given, the rigid body rolls without sliding. Hence, $a=\alpha R$
and friction$\left(f\right)=0$

Torque about centre of mass
$\tau =-F\times x=-I_{\text{cm}}\alpha$ ... (1)
[-ve indicates clockwise direction]
From $\sum F_x=ma$
$F=ma$ ... (2)
From eqn. (1) and (2)
$max=I_{\text{cm}}\alpha =I_{\text{cm}}\frac{a}{R}$ $(\because a=\alpha R)$
$\Rightarrow x=\frac{I_{\text{cm}}}{mR}$

Note: For pure rolling, if any friction is required, it will be static friction. It may be zero, in backward direction or forward direction depending on the value of $x$. If $F$ is below the point $P$, then friction force will act in backward direction and if it is above the point $P$ friction force will act in forward direction.