A block of mass m is tied to a light string wound around a cylinderical wheel th

Question

A block of mass m is tied to a light string wound around a cylinderical wheel that has a mass M and radius R. The pulley bearing is frictionless, and the string does not slip on the rim. Consider that the initial state of the system is when the hanging mass is at height H with respect to the ground and moving with velocity vi.

a) Wrrite the total energy of the system in the initial state expressing kinetic and potential energy specifically for each object.

b) Write the total energy of the system for the final position, after mass m has fallen a distance H, expressing kinetic and potential energy specifically for each object.

c) Write the equation that connects the angular velocity of the wheel to the linear velocity of the hanging mass for both the initial and final positions.

d) Write the conservation of energy between the initial and final states of the system. The equation should be in terms of I, vi, vf, m, g, R, and H.

e) Using the equation from the previous part d, find the moment of inertia of the wheel.

Explanation / Answer

a) potential energy of the block(V)= mgH

kinetic energy of the block(K)=m*vi2/2

Rotational kinetic energy of the wheel(Kr)= I*w2/2 where I- moment of inertia=M*R2 and angular velocity(w)=vi/R

Kr=M*vi2/2

tootal energy=K + Kr + V

b) Potential energy of the block= 0

Kinetic energy of the block(K')= m*vf2/2

Rotational kinetic energy of the wheel(Kr')=M*vf2/2

Total energy= K' + Kr'

c) vi= R*wi ; wi- initial angular velocity

vf= R*wf ; wf- final angular velocity

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