The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle without slipping and slipping down the incline without rolling is [2014]

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Solution

Q.
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 [2014]

1 5 : 7

2 2 : 3

3 2 : 5

4 7 : 5

Ans.
(1)

Acceleration of the solid sphere slipping down the incline without rolling is

   $a_{slipping} = g \sin θ$ ...(i)

Acceleration of the solid sphere rolling down the incline without slipping is

   $a_{rolling} = \frac{g \sin θ}{1 + \frac{k^{2}}{R^{2}}} = \frac{g \sin θ}{1 + \frac{2}{5}} = \frac{5}{7} g \sin θ$ ..(ii)

   $(∵ For solid sphere, \frac{k^{2}}{R^{2}} = \frac{2}{5})$

Divide eqn. (ii) by eqn. (i), we get

$\frac{a_{rolling}}{a_{slipping}} = \frac{5}{7}$

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