Question

Q. A sphere of mass m rolls without slipping on an inclined plane of inclination theta. Find the linear acceleration of the sphere and the force of friction acting on it. What should be the minimum coefficient of static friction to support pure rolling ?

Answer 1

Answer:

Linear acceleration, friction force and coefficient of static friction for rolling of sphere is given as

$a = \frac{5gsin\theta}{7}$

$F_f = \frac{2}{7}mg sin\theta$

$\mu = \frac{2tan\theta}{7}$

Explanation:

For inclined plane of inclination theta we can say

$mg sin\theta - F_f = ma$

also for torque equation we have

$F_f R = I \alpha$

here we know

$I = \frac{2}{5}mR^2$

$\alpha = \frac{a}{R}$

so we have

$F_f = \frac{2}{5}m a$

now we have

$mg sin\theta = ma + \frac{2}{5}ma$

$a = \frac{5gsin\theta}{7}$

also from above equation we have

$F_f = \frac{2}{5}ma$

$F_f = \frac{2}{7}mg sin\theta$

now for friction coefficient we will have

$F_f = \mu mg cos\theta$

so we have

$\mu = \frac{2tan\theta}{7}$

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Topic : Rolling on inclined plane

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