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 74.0 kg that rests on a platform. The cylinder has a mass of 205 kg and a radius of 0.370 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?

Explanation / Answer

Part A)

For the hanging mass, the sum of forces is

mg - T = ma

(74)(9.8) - T = 74(a)

T = 725.2 - 74a

For the pulley

torque = I(alpha)

Torque = Tr

I = .5mr2 and alpha = a/r, so...

Tr = .5mr2a/r which simplifies to

T = .5ma

Set the T's equal

.5(205)a = 725.2 - 74a

a = 4.11 m/s2

alpha = a/r = 4.11/.37

alpha = 11.1 rad/s2

Part B)

Theta = wot + .5(alpha)(t2)

Theta = (.5)(11.1)(5)2

Theta = 138.8 rad

For rev, divide by 2pi

That is 22.1 revs

Part C)

Length = 22.1 circumferences

22.1(2)(pi)(.37) = 51.4 m

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