Question

A sphere can roll on a surface inclined at an angle θ if the friction coefficient is more than $\frac{2}{7}g\mathrm{tan\theta }$. Suppose the friction coefficient is $\frac{1}{7}g\mathrm{tan\theta }$. If a sphere is released from rest on the incline,
(a) it will stay at rest
(b) it will make pure translational motion
(c) it will translate and rotate about the centre
(d) the angular momentum of the sphere about its centre will remain constant.

Answer 1

(c) it will translate and rotate about the centre

The given coefficient of friction ($\frac{1}{7}g\mathrm{tan\theta }$) is less than the coefficient friction ($\frac{2}{7}g\mathrm{tan\theta }$) required for perfect rolling of the sphere on the inclined plane. Therefore, sphere may slip while rolling and it will translate and rotate about the centre.