(Solved): A hanging weight, with a mass of \( m_{1}=0.350 \mathrm{~kg} \), is attached by a rope to a block w ...

A hanging weight, with a mass of \( m_{1}=0.350 \mathrm{~kg} \), is attached by a rope to a block with mass \( m_{2}=0.840 \mathrm{~kg} \) as shown in the figure below. The rope goes over a pulley with a mass of \( M=0.350 \mathrm{~kg} \). The pulley can be modeled as a hollow cylinder with an inner radius of \( R_{1}=0.0200 \mathrm{~m} \), and an outer radius of \( R_{2}=0.0300 \mathrm{~m} \); the mass of the spokes is negligible. As the weight falls, the block slides on the table, and the coefficient of kinetic friction between the block and the table is \( \mu_{k}=0.250 \). At the instant shown, the block is moving with a velocity of \( v_{j}=0.820 \mathrm{~m} / \mathrm{s} \) toward the pulley. Assume that the pulley is free to spin without friction, that the rope does not stretch and does not slip on the pulley, and that the mass of the rope is negligible. (a) Using energy methods, find the speed of the block (in \( \mathrm{m} / \mathrm{s} \) ) after it has moved a distance of \( 0.700 \mathrm{~m} \) away from the initial position shown. X \( \mathrm{m} / \mathrm{s} \) (b) What is the angular speed of the pulley (in rad/s) after the block has moved this distance? \( X \mathrm{rad} / \mathrm{s} \)