A light rope is wrapped several times around a large wheel with a radius of 0.40

Question

A light rope is wrapped several times around a large wheel with a radius of 0.400 m. The wheel rotates in frictionless bearings about a stationary horizontal axis, as shown in the figure. The free end of the rope is tied to a suitcase with a mass of 15.0 kg. The suitcase is released from rest at a height of 4.00 m above the ground. The suitcase has a speed of 3.50 m/s when it reaches the ground, Calculate the moment of inertia of the w heel, What was the angular acceleration of the wheel and the linear acceleration of the suitcase during the motion?

Explanation / Answer

a)   Here,   loss in potential energy of suitcase = gain in kinetic energy (rotational + linear)

=> 15 * 9.8 * 4   = 1/2 * 15 * 3.502   + 1/2 * I * w2

=>    15 * 9.8 * 4   = 1/2 * 15 * 3.502   + 1/2 * I * (v/r)2

=>    15 * 9.8 * 4   = 1/2 * 15 * 3.502   + 1/2 * I * (3.50/0.400)2

=>    I   =   12.96 kg-m2

=> moment of inertia of wheel = 12.96 kg-m2

b) linear acceleration of suitcase = 3.50 * 3.50/(2 * 4)

                                                      = 1.531 m/sec2

=> angular acceleration of wheel = 1.531/0.4

                                                         = 3.827 rad/sec2

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