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) $5 : 21$
Concept. A force at the top of a rolling body: write $F+f=ma$ (friction $f$ unknown) and $(F-f)R=I\alpha$ with $a=\alpha R$. Adding gives $a=\dfrac{2F}{m(1+k^2/R^2)}$.
Step 1. Sphere A ($5m$, $k^2/R^2=\tfrac25$). $a_A=\dfrac{2F}{5m\cdot\tfrac75}=\dfrac{2F}{7m}$.
Step 2. Shell B ($m$, $k^2/R^2=\tfrac23$). $a_B=\dfrac{2F}{m\cdot\tfrac53}=\dfrac{6F}{5m}$.
Step 3. Ratio. $\dfrac{a_A}{a_B}=\dfrac{2/7}{6/5}=\dfrac{10}{42}=\dfrac5{21}$.
Answer. Option A.
Solution by Sreeraj P, M.Sc Physics
