$[0/1$ Points$]$

SERPSE$1010\mathrm{.}8\mathrm{.}0$P$\mathrm{.}025.$

A hanging weight, with a mass of $m_1=0\mathrm{.}350kg,$ is attached by a rope to a block with mass $m_2=0\mathrm{.}800kg$ as shown in the figure below. The rope goes over a pulley with a mass of $M=0\mathrm{.}350kg.$ The pulley can be modeled as a hollow cylinder with an inner radius of $R_1=0\mathrm{.}0200m,$ and an outer radius of $R_2=0\mathrm{.}0300m$; 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 ${}_k=0\mathrm{.}250.$ At the instant shown, the block is moving with a velocity of $v_j=0\mathrm{.}820\frac{m}{s}$ 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 $\frac{m}{s})$ after it has moved a distance of $0\mathrm{.}700$ m away from the initial position shown.

$0\mathrm{.}8$:

Use conservation of energy, treating the universe as the system of interest. What types of energy are changing? How do you relate the kinetic energy of the pulley to its moment of inertia and angular speed? What is the moment of inertia of a hollow cylinder? How is the angular speed related to the linear speed of the block? What is the internal energy change of the system, and how is it related to friction? $\frac{m}{s}$

$($b$)$ What is the angular speed of the pulley $($in rad$/$s$)$ after the block has moved this distance?

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How is angular speed related to linear speed? What is the final speed of the block found in part $($a$)?$ rad$/$s