A solid cylinder is mounted above the ground with its axis of rotation oriented

Question

A solid cylinder is mounted above the ground with its axis of rotation oriented horizontally. A rope is wound around the cylinder and its free end is attached to a block of mass 92.5 kg that rests on a platform. The cylinder has a mass of 215 kg and a radius of 0.460 m. Assume that the cylinder can rotate about its axis without any friction and the rope is of negligible mass. The platform is suddenly removed from under the block. The block falls down toward the ground and as it does so, it causes the rope to unwind and the cylinder to rotate. (a) What is the angular acceleration of the cylinder? rad/s2 (b) How many revolutions does the cylinder make in 5 s? rev (c) How much of the rope unwinds in this time interval? m

Explanation / Answer

Given,

m = 92.5 kg ; M = 215 kg ; r = 0.46 m ;

a)Let T be the tension in the cord.

mg - T = ma

T = mg - ma

we know that,

a = alpha r

a = alpha x 0.46

T = mg - m alpha r

T = 92.5 x 9.8 - 92.5 x alpha x 0.46 = 906.5 - 42.55 alpha

T = 906.5 - 42.55 alpha

we know that, torque is:

Torque = F x r = I alpha ; I is the moment of inertia

Force here is tension in the cord

T r = 1/2 m r^2 alpha

T 0.46 = 1/2 x 215 x 0.46^2 x alpha

T = 49.45 alpha

49.45 alpha = 906.5 - 42.55 alpha

alpha = 9.85 rad/s^2

Hence, alpha = 9.85 rad/s^2

b)t = 5 s

theta = 1/2 alpha t^2

theta = 0.5 x 9.85 x 5^2 = 123.13 rad

N = 123.13/2 x 3.14 = 19.61 rev

Hence, N = 19.61 rev

c)S = ut + 1/2 at^2

S = 0 + 1/2 x 9.85 x 0.46 x 5^2 = 56.67 m

Hence, S = 56.67 m

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