A pulley of moment of inertia I and radius r is fixed to a wall. A rope is wrapp

Question

A pulley of moment of inertia I and radius r is fixed to a wall. A rope is wrapped around it and tied to a block of mass m. The block is initially held at rest, and then it is released. The block drops a distance h. The rope does not slip on the pulley, and the magnitude of the acceleration due to gravity is g. What is the work done by gravity on the system? You know I, r, m, h, and g.

What is the change in potential energy of the system?

What is the change in kinetic energy of the system?

Write the equation that expresses the no-slip condition – the relationship between the linear and angular speeds.

What is the speed v of the block in terms of I, r, m, h, and g?

What is the angular speed ? of the pulley in terms of I, r, m, h, and g?

Explanation / Answer

Part A

work done by gravity W = force * displacement = mgh

Part B

change in potential energy delta U = -mgh

part C

Change in kinetic energy

delta E = work done = mgh

part D

In no slip condition

linear speed = angular speed*radius

V = wr

Part E

By work energy theorm

workdone = cahnge in translational KE and Rotational KE

mgh= 1/2mv^2 +1/2Iw^2

mgh = 1/2mv^2 + 1/2I(V/r)^2

mgh = 1/2mv^2 + 1/2(I/r^2)v^2

mgh = 1/2v^2(m+I/r^2)

v = 2mgh/(m+I/r^2)

Part F

w = v/r

w = 2mgh/(m+I/r^2)*r

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