The figure below shows a pulley in the form of a uniform disk with a rope hangin

Question

The figure below shows a pulley in the form of a uniform disk with a rope hanging over it. The circumference of the pulley is 1.4 m and its mass is 2.1 kg. The rope is 8.8 m long and its mass is 4.9 kg. At the instant shown in the figure, the system is at rest and the difference in height of the two ends of the rope is 0.70 m.

What is the angular speed of the pulley when the difference in height between the two ends is 7.2 m?

Obtain an expression for the angular momentum of the system as a function of time while neither end of the rope is above the center of the pulley. There is no slippage between rope and pulley. (Use the following as necessary: t.)

Explanation / Answer

Info given-

Circumference of pulley = 1.4 m

Radius of pulley = 1.4 / 2 * 3.14 = 0.22 m.

Mass of pulley = 2.1 kg

Mass of rope = 4.9 kg

Force on both sides of the pulley will be expressed as-

Mp.g = [Mr + Mp]a

2.1 * 9.8 = [2.1 + 4.9]a

a= 2.94 m/s2.

When the pulley moves with a = 2.94 m/s2, distance moved = (7.2 – 0.7) = 6.5 m. But since the rope has moved on both the sides, distance would be calculated as half = s = 6.5 / 2= 3.25 m.

Now using Newton’s equation of motion-

v2 – u2 = 2gs = [2 * 9.8 * 3.25] – 0 = 63.7

v = 7.98 m/s.

Angular speed = v/r = 7.98 / 0.22 = 36.27 rad/sec.

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