Physics - Rotation Question with Solution | TestHub

Let $a_c$ be the acceleration of the center of mass of the sphere and $\alpha$ be its angular acceleration. For rolling without slipping, $a_c=R\alpha$. Let the applied force $F$ be to the right. Assume the friction force $f$ acts to the left. Applying Newton's second law for translational motion: $F-f=M{a}_c$. Applying Newton's second law for rotational motion about the center of mass: The force $F$ creates a clockwise torque $FR$, and the friction force $f$ creates a counter-clockwise torque $fR$. So, $FR-fR=I_{sphere}\alpha$. For a solid sphere, $I_{sphere}=\frac{2}{5}M{R}^2$. Thus, $R(F-f)=\frac{2}{5}M{R}^2\alpha ⟹F-f=\frac{2}{5}MR\alpha$. Substituting $\alpha =a_c/R$ into this equation gives $F-f=\frac{2}{5}M{a}_c$. We now have two equations: $F-f=M{a}_c$ and $F-f=\frac{2}{5}M{a}_c$. For both equations to hold, it must be that $M{a}_c=\frac{2}{5}M{a}_c$, which implies $a_c=0$. If $a_c=0$, then from $F-f=M{a}_c$, we get $F-f=0$, so $f=F$. The magnitude of the friction force is $F$, and its direction is to the left.