First slide

Rolling motion

Question

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. In both cases

Moderate

A

the speed and time of descend will be same
B

the speed will be same but time of descend will be different
C

the speed will be different but time of descent will be same.
D

speed and time of descent both are different.

Solution

Speed of the rolling body at the bottom of inclined plane is
$v = \sqrt{\frac{2 gh}{1 + \frac{k^{2}}{R^{2}}}}$
where h is height of the inclined plane, k and R be radius of gyration and radius of the body respectively.
For solid sphere, $\frac{k^{2}}{R^{2}} = \frac{2}{5}$
$v = \sqrt{\frac{2 gh}{1 + \frac{2}{5}}} = \sqrt{\frac{10}{7} gh}$
As h is same in both the cases, therefore speed will be same in both cases.
Time of descent,
t = $\frac{1}{sinθ} \sqrt{\frac{2 h}{g} (1 + \frac{k^{2}}{R^{2}})}$
Where $θ$ is angle of inclination?
For solid sphere, $\frac{k^{2}}{R^{2}} = \frac{2}{5}$
$t = \frac{1}{sinθ} \sqrt{\frac{14 h}{5 g}}$
As h is same but $θ$ is different in both the cases, hence time of descend will be different in both the cases.

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