Question

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

$a$ is linear acceleration, $\alpha$ is angular acceleration and $I_{com}$ is the rotational inertia of the body about its COM.

• A. $\frac{I_{com}}{2mR}$
• B. $\frac{I_{com}}{mR}$
• C. $\frac{I_{com}}{4mR}$
• D. $\frac{2I_{com}}{mR}$

Answer 1

The correct option is B $\frac{I_{com}}{mR}$


Taking torque about point of contact $O$.

Then,

$(I_{com}+m{R}^2)\alpha =(R+x)F$

$\Rightarrow (I_{com}+m{R}^2)\alpha =(R+x)ma$

$\Rightarrow (I_{com}+m{R}^2)\alpha =(R+x)m\left(\alpha R\right)$

$\Rightarrow x=\frac{I_{com}}{mR}$