Question
Asked Nov 13, 2019
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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 1.7 m and its mass is 2.9 kg. The rope is 9.2 m long and its mass is 5.2 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 0.85 m.

(a) What is the angular speed of the pulley when the difference in height between the two ends is 7.2 m?
(b) 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.)

d
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d -

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Expert Answer

Step 1

Solution:

Radius r of the pulley can be calculated as,

 

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27r 1.7m 1.7m r = 2T = 0.27m

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Step 2

The mass per unit length μ of the rope of mass m and length l is given as,

 

μ = m/l

 

Substitute 5.2 kg for m and 9.2 m for l and solve for μ as,

 

μ = 5.2 kg/9.2 m

 

   = 0.565 kg/m

 

Let the length of the portion of the pulley on the left side be denoted as i and on the right side be denoted as j.

 

It can be seen that

i + 0.85 = j

 

Also,

 

i + j = 9.2

 

From the above two equations,

 

The value of i and j can be computed as,

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2i0.85 9.2 i =4.175m i+0.85= j 5.025m

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Step 3

Corresponding weights on the left...

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m, (4.175m)(0.565kg/m) 2.36kg m, (5.025m)(0.565kg/m) 2.84kg

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