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Question 7.21:
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?
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.