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A 6-kg block (m1), initially at rest on a rough shelf, is connected to a 4-kg block (m2) that hangs by an inextensible string of negligible mass passing over a pulley. The uniform disk-shaped pulley has a mass of 3.0-kg and radius 20 cm. The string does not slip on the pulley and causes the pulley to rotate about a fixed horizontal axle through its center of mass. The 6-kg block, which is not attached to the spring, is pushed to compress the spring through a displacement of 40 cm from its equilibrium position. The force constant of the spring is 500 N/m and the blocks move with a uniform acceleration after being released. The coefficient of kinetic friction between the 6-kg block and the shelf surface is 0.20. After the block is released, calculate the speed of the blocks when the spring first returns to its equilibrium length. [v = 3.0 m/s]
A block of mass m1 = 6 kg on a rough 30°-inclined plane is connected to a 4-kg mass (m2) by a string of negligible mass passing over a pulley shaped like a ring. The 2-kg pulley has radius 20 cm and rotates about its symmetry axis of rotation. The string causes the blocks and the pulley to rotate without slipping and without friction. The 6-kg block (m1) on the 30°-slope is initially pressed against a spring near the bottom of a long rough incline, compressing the spring by 50 cm. The spring is not attached to the block and has a spring constant is 500 N/m. When the system is released, the spring returns to its equilibrium length as it projects the 6-kg block (m1) toward the top of the incline. Assume that the spring just loses contact with the block (m1) at the instant it returns to its equilibrium length. The coefficient of kinetic friction between the block (m1) and the surface of the incline is 0.2. By considering the conservation of energy and any other suitable methods:
(a) Calculate the speed of the blocks at the instant the spring first returns to its equilibrium length. [v = 3.2 m/s]
(b) What is the total angular momentum of the system when the spring first returns to its equilibrium length? [7.7 kg.m2/s]
(c) By considering the total angular momentum of the system, find the rate of change of the angular momentum of the system after the blocks lose contact with the spring. [-0.08 m.N]
(d) What is the net torque causing the angular acceleration of the system after the blocks lose contact with the spring? [-0.08 m.N]

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