Question
A solid sphere (A) of mass, 5 m and 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 $\_\_\_\_$ .
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Solution
Sol. [Key Concept: Force at the top gives torque; for rolling, $2F = (m + \frac{I}{R^{2}})a$: the solid sphere ($I = \frac{2}{5}\ {mR}^{2}$) and the shell ($I = \frac{2}{3}\ {mR}^{2}$) give $a_{A}:a_{B} = 5:21$.]
$${F + f = 5\text{ }ma_{A} }{F(R) - f(R) = \frac{2}{5}(5\text{ }m)R^{2}.\alpha }{\alpha = R\alpha }{2\text{ }F = 7{ma}_{A} }{a_{A} = \frac{2\text{ }F}{7\text{ }m} }$$
$${F + f = {ma}_{B} }{F(R) - f(R) = \frac{2}{3}{mR}^{2} \times \alpha }{a = R\alpha}$$
$$2\text{ }F = \frac{5{ma}_{B}}{3}$$
$$\frac{a_{A}}{a_{B}} = \frac{2\text{ }F}{7\text{ }m} \times \frac{5\text{ }m}{6\text{ }F} = \frac{5}{21} $$
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