A solid cylinder and a solid sphere having the same mass and radius roll down on the same smooth inclined plane. The ratio of the acceleration of the cylinder ( a c ) to that of the sphere ( a s )

Question

A solid cylinder and a solid sphere having the same mass and radius roll down on the same smooth inclined plane. The ratio of the acceleration of the cylinder ( a c ​ ) to that of the sphere ( a s ​ ) is

TestHub · Accepted Answer

The acceleration of a body rolling down an inclined plane is given by $a = \frac{g sin θ}{1 + \frac{I}{M R ^{2}}}$ .

For a solid cylinder, $I_{c} = \frac{1}{2} M R^{2}$ .

$a_{c} = \frac{g sin θ}{1 + \frac{1}{2}} = \frac{2}{3} g sin θ$

For a solid sphere, $I_{s} = \frac{2}{5} M R^{2}$ .

$a_{s} = \frac{g sin θ}{1 + \frac{2}{5}} = \frac{5}{7} g sin θ$

The ratio of the acceleration of the cylinder to that of the sphere is:

$\frac{a _{c}}{a _{s}} = \frac{\frac{2}{3} g sin θ}{\frac{5}{7} g sin θ} = \frac{14}{15}$

Answer: $\frac{14}{15}$

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