You are pulling on a rope attached to the outer radius of a pulley with a moment

Question

You are pulling on a rope attached to the outer radius of a pulley with a moment of inertia I = 2.85 kg-m^2 as shown. The tension force of you on the pulley is 266.3 N. A pack of mass 23.4 kg is attached to the pulley's inner radius as shown. If the inner radius of the pulley is 0.43 m and the outer radius is 0.66 m, what is the magnitude of the linear acceleration of the pack? If instead of using a pulley with two radii, you were to replace the two radii pulley with a simple pulley of one radius: how would using the simple pulley instead of the two radii pulley affect the difficulty of holding the pack at rest over the edge? It will be easier to hold the pack. Not enough information to tell. There will be no difference. It will be harder to hold the pack.

Explanation / Answer

Let Tr be the tension in the rope, Tm be the tension in the cors connecting inner radius Ri and pack, T be the torque and Ro be the outer radius.

Torque T will be

T = Tr Ro - Tm Ri

we know that, T = I alpha => alpha = T/I

alpha = (Tr Ro - Tm Ri)/I

Tm = m (g + a)

alpha = [Tr Ro - m(g + a) Ri]/I

alpha x I = Tr Ro - m g Ri + ma Ri

a = alpha Ri

alpha I = Tr Ro - m g Ri - m alpha Ri^2

alpha = (Tr Ro - m g Ri)/(1 + m Ri^2)

linear acceleration would be, a = Ri alpha

a = Ri [(Tr Ro - m g Ri)/(1 + m Ri^2)]

a = 0.43 [ 266.3 x 0.66 - 23.4 x 9.8 x 0.43]/(1 + 23.4 x 0.43^2] = 0.271 m/s^2

Hence, a = 0.271 m/s^2

b)It will be harder to hols the pack

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