3. (6 pts) (Essentially 9.80) In the following system, the cylinder and the pull

Question

3. (6 pts) (Essentially 9.80) In the following system, the cylinder and the pulley turn without friction about stationary horizontal axles that pass through their centers. A light rope is wrapped around the cylinder, passes over the pulley, and has a box, mB, suspended from its free end. There is no slipping between the rope and the pulley surface. The uniform cylinder has mass mc and radius Rc. The pulley is a uniform disk with mass mP and radius RP. The box is released from rest and descends as the rope unwraps from the cylinder. Find an expression for vf, the speed of the box when it has fallen a distance d. Pulley Cylinder Box

Explanation / Answer

Applying energy conservation,

initial PE+ KE = final PE +KE

m g h + 0 = 0 + (mB v^2 /2 ) + ( Ip wp^2 /2 ) + (Ic wc^2 / 2)

2 mB g h = mB v^2 + (mp Rp^2 /2 )(v/Rp)^2 + (mC Rc^2 / 2)(v / Rc)^2

4 mB g h = (2mB + mp + mC) v^2

v = sqrt[ 4 mB g h / (2mB + mp + mC )]

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