The pulley in Fig. P9 .83 has radius R and a moment of...

The pulley in Fig. $\mathrm{P9} .83$ has radius $R$ and a moment of inertia $I .$ The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block $A$ and the tabletop is $\mu_{\mathrm{k}}$ . The system is released from rest, and block $B$ descends. Block $A$ has mass $m_{A}$ and block $B$ has mass $m_{B}$ . Use energy methods to calculate the speed of block $B$ as a function of the distance $d$ that it has descended.

Key Concepts

Work Done by Friction

Friction, a non-conservative force, performs work that dissipates mechanical energy from the system. In this context, the work done by kinetic friction on the sliding block reduces the available energy that is converted into kinetic energy, and must be subtracted when applying the energy conservation principle.

Conservation of Energy

This principle states that the total energy in a closed system remains constant, but energy can change forms. In problems like this one, gravitational potential energy, translational kinetic energy, rotational kinetic energy, and work done by friction all need to be accounted for when using energy methods to determine the final speed of the system.

Rotational Kinetic Energy

Rotational kinetic energy is the kinetic energy due to an object's rotation. When a pulley is involved, its moment of inertia and the angular velocity (related to the linear speed by the non-slip condition) contribute an additional energy term that must be included alongside translational kinetic energies.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion. For the pulley, it is a crucial parameter because it determines how much energy is needed to achieve a certain angular velocity. This factor directly influences the distribution of the system's energy between rotational and translational forms.

No-Slip Condition

The no-slip condition ensures that the rope remains in contact with the pulley without sliding. This condition provides a kinematic relation between the linear speed of the block and the angular speed of the pulley, allowing the conversion between linear and rotational kinetic energies.

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