Physics - Rotation Question with Solution | TestHub

Let $a$ be the acceleration of the center of mass and $\alpha$ be the angular acceleration. Let $f$ be the friction force acting on the sphere. Translational equation: $F-f=Ma$ (1)

Rotational equation about the center of mass: $\tau =I\alpha$. The force $F$ acts at a distance $R$ from the center, creating a torque $FR$. The friction force $f$ also creates a torque $fR$. Both torques are in the same direction to cause rolling. For a solid sphere, $I=\frac{2}{5}M{R}^2$. So, $FR+fR=I\alpha =(\frac{2}{5}M{R}^2)\alpha$ (2) For rolling without slipping, $a=R\alpha$, so $\alpha =a/R$. Substitute $\alpha$ into (2):$FR+fR=\frac{2}{5}M{R}^2(a/R)⟹F+f=\frac{2}{5}Ma$ (3) Now we have two equations: (1) $F-f=Ma$ and (3) $F+f=\frac{2}{5}Ma$. Add (1) and (3): $(F-f)+(F+f)=Ma+\frac{2}{5}Ma$ $$2F=(1+\frac{2}{5})Ma=\frac{7}{5}Ma$$ $$a=\frac{10F}{7M}$$