First slide

Rolling motion

Question

A ring, cylinder and solid sphere are placed on the top of a rough incline on which the sphere can just roll without slipping. When all of them are released at the same instant from the same position, then

Moderate

A

all of them reach the ground at the same instant
B

the sphere reaches first and the ring at the last
C

the sphere reaches first and the cylinder and ring reach together
D

none of the above

Solution

For a body to roll without slipping on an incline of angle $θ$ :
$μ \geq \frac{\tan θ}{1 + \frac{m R^{2}}{I}}$
I is minimum for the sphere, hence $μ$ required is least for the sphere. Here it is given that $μ$ is just sufficient to roll the sphere purely. It means the ring and the cylinder will not roll purely, so kinetic friction will act on them. Ultimately, friction acting on each of the three will be $μ$ mg cos $θ$ and each of them will have the same acceleration. Hence, all of them reach at the same time.

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