A solid sphere ($A$) of mass $5m$ and a spherical shell ($B$) of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of $A$ and $B$, they start rolling without slipping with an acceleration of $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is ______.
Answer: (A) $5 : 21$
Take torques about the contact point (instantaneous axis). The force $F$ at the top has arm $2R$:
$$F(2R) = (I_{cm} + MR^2)\frac{a}{R} \;\Rightarrow\; a = \frac{2F}{M(1 + k)},\quad k = \frac{I_{cm}}{MR^2}$$
Solid sphere ($M = 5m$, $k = \frac{2}{5}$): $a_A = \dfrac{2F}{5m \times \frac{7}{5}} = \dfrac{2F}{7m}$.
Shell ($M = m$, $k = \frac{2}{3}$): $a_B = \dfrac{2F}{m \times \frac{5}{3}} = \dfrac{6F}{5m}$.
$\dfrac{a_A}{a_B} = \dfrac{2/7}{6/5} = \dfrac{10}{42} = \dfrac{5}{21}$.
Solution by Sreeraj P, M.Sc Physics