A cylinder of mass M , length L and radius R has two cords wound around it with their ends attached to the ceiling, as shown in the figure. The cylinder is held horizontally with the two cords vertical. When the system is released from rest, the linear acceleration of the cylinder as it falls will be

Step-by-step solution

The cylinder rotating under gravity has both translational and rotational motions. Let v be the linear velocity of its centre of mass and ω its angular velocity about the axis of rotation, then in descending a distance h it will lose PE = Mgh while the gain in KE = 1 2 I ω 2 + 1 2 M v 2 , so Mgh = 1 2 I ω 2 + 1 2 Mv 2 Here I = 1 2 MR 2 and v = R ω so Mgh = 1 2 Mv 2 + 1 2 1 2 MR 2 v 2 R 2 so v 2 = 4 3 g h Now differentiating both sides with respect to time 2v dv dt = 4 3 g dh dt or a = 2 3 g as