Question

A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.

(a) How far will the cylinder go up the plane?

(b) How long will it take to return to the bottom?

Answer 1

A solid cylinder rolling up an inclination is shown in the following figure.

Initial velocity of the solid cylinder, v = 5 m/s

Angle of inclination, θ = 30°

Height reached by the cylinder = h

(a) Energy of the cylinder at point A:

Energy of the cylinder at point B = mgh

Using the law of conservation of energy, we can write:

Moment of inertia of the solid cylinder,

In ΔABC:

Hence, the cylinder will travel 3.82 m up the inclined plane.

(b) For radius of gyration K, the velocity of the cylinder at the instance when it rolls back to the bottom is given by the relation:

The time taken to return to the bottom is:

Therefore, the total time taken by the cylinder to return to the bottom is (2 × 0.764) 1.53 s.