An Atwoodís Machine consists of two weights connected by a string that is looped

Question

An Atwoodís Machine consists of two weights connected by a string that is looped over a pulley. In our previous investigations, we considered the pulley to be very light so that its mass could be ignored. Assume now that the pulleyís mass is important. Suppose that the weight on the left has a mass of 3.0 kg, the weight on the right has a mass of 2.0 kg, and the pulley has a mass of 10.0 kg, and a radius of 5.0 cm. The system is released from rest.

(a) What is the angular acceleration of the pulley if it is shaped like a hoop (or a bicycle wheel), with all of its mass on the rim? What is the tension in the string?

(b) What is the angular acceleration of the pulley if it is a solid cylinder? What is the tension in the string?

Explanation / Answer

Applying Fnet = m a on right mass.

3g - T1 = 3 a

on left mass: T2 - 2g = 2 a

(A) for hoop:

r(T1 - T2) = (m r^2) (a /r)

T1 - T2 = 10 a

adding all three equations,

3g - 2g = (3 + 2+ 10) a

a = 0.654 m/s^2

alpha = a / r = 13.1 rad/s^2 ..........Ans

T1 = 3 ( g - a) = 27.5 N .... tension in right side string

T2 = 2(g + a) = 20.9 N ...Tensionin left side string

(b) for solid cylinder:

T1 - T2 = 5 a

a = 0.981 m/s^2

alpha = a/r = 19.62 rad/s^2 ...Ans

tension in right side string = 3 (g - a) = 26.5 N

tension in left side string = 2(g + a) = 21.6 N

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