First slide

Rolling motion

Question

The ratio of the accelerations for a solid sphere (mass m and radius R ) rolling down an incline of angle $θ$ without slipping and slipping down the incline without rolling is

Moderate

A

5 : 7
B

2 : 3
C

2 : 5
D

7 : 5

Solution

Acceleration of the solid sphere slipping down the incline without rolling is
$a_{slipping} = g \sin θ$
Acceleration of the solid sphere rolling down the incline without slipping is
$a_{rolling} = \frac{g \sin θ}{1 + \frac{k^{2}}{R^{2}}} = \frac{g \sin θ}{1 + \frac{2}{5}} (∵ For solid sphere, \frac{k^{2}}{R^{2}} = \frac{2}{5})$
$= \frac{5}{7} g \sin θ . . . . . (i i)$
Divide eqn. (ii) by eqn. (i), we get
$\frac{a_{rolling}}{a_{slipping}} = \frac{5}{7}$

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