An Atwood’s machine with a pulley in the shape of a uniform disk of mass M and r

Question

An Atwood’s machine with a pulley in the shape of a uniform disk of mass M and radius R is in an elevator having vertical acceleration A relative to the ground. Let positive A correspond to the elevator accelerating upward. Mass mA hangs on the left side of the pulley while mass mB hangs on the right. The rope connecting masses mA and mB is massless. The pulley is suspended from the ceiling of the elevator by a spring of spring constant k and natural length l.

(a) Consider the special case mA = mB. What would you expect the length of the spring to be in the following limits:

i. A ? 0

ii. A ? ?g

iii. A ? g

Justify your answers using physical reasoning and minimal math if possible.

(b) When mA is not equal to mB but A = 0, would you expect the tension in the spring to be less than, equal to, or greater than the weight of the pulley plus the weight of the masses hanging from the pulley? Justify your answers using physical reasoning and minimal math if possible.

(c) Determine a general expression for the length of the spring in terms of the given variables. Showing work

Explanation / Answer

a)    i. A 0

   length of the spring =   l + [(2mA + M) * g]/k

ii. A g

length of the spring =   l + [(2mA + M) * g]/2k

iii. A g

length of the spring =   l + [2(2mA+ M) * g]/k

b)    The tension in the spring would be equal to, the weight of the pulley plus the weight of the masses hanging from the pulley .

c)   As, Force in spring , F = kx

=> general expression for the length of the spring =   natural length + stretched length

                                                                           =   l + [(mA + mB + M) * (g + A)]/k

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