A uniform solid sphere of mass $M$ and radius $R$ is placed on a rough horizontal surface. A horizontal force $F$ is applied at a height $h$ above the center of the sphere. For the sphere to roll without slipping immediately after the force is applied, the value of $h$ must be:

Solution

1. The equations of motion are $F=Ma$ and $Fh+fR=I\alpha$, where $f$ is friction and $I=\frac{2}{5}M{R}^2$.
2. For pure rolling, $a=\alpha R$.
3. Substituting $f=\mu N$ is not needed; we use the constraint $a=\alpha R$ to find $f=F(\frac{2R^2}{5R^2}-\frac{h}{R})/(1+\frac{2}{5})$.
4. For the sphere to roll without slipping from rest, the friction must satisfy the constraint $a=\alpha R$, leading to $h=\frac{2}{5}R$.
5. Hence the answer is (B).