In the figure below the hanging object has a mass of m1 037

In the figure below, the hanging object has a mass of m_1 = 0.375 kg; the sliding block has a mass of m_2 = 0.870kg; and the pulley is a hollow cylinder with a mass of M = 0.350kg, an inner radius of M = 0.020 0m, and an outer radius of R_2 = 0.030 0m. Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is mu_k = 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of v_i = 0.820m/s toward the pulley when it passes a reference point on the table. Use energy methods to predict its speed after it has moved to a second point, 0.700m away. Find the angular speed of the pulley at the same moment.

Solution

v_f = sqrt [v_i² + (m₁ g h - mu_k m₂ g) / (1/2(m₁ + m₂) + 1/2M(1 + R₁²/R₂²))]

= sqrt [0.820² + (0.375*9.8*0.700 - 0.250*0.870*9.8) / (1/2(1.245) + 1/2(0.350)(1 + 0.02²/0.03²))]

= sqrt [0.820² + (0.441) / (0.6225 + 0.2527)]

= 1.08 m/s

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w_f = v_f / R₂

= 1.08 / 0.030

= 36.15 rad/s