The pulley has a weight of 8 lb and may be treated as a thin...

The pulley has a weight of 8 lb and may be treated as a thin disk. A cord wrapped over its surface is subjected to forces $T_{A}=4 \mathrm{lb}$ and $T_{B}=5 \mathrm{lb} .$ Determine the angular velocity of the pulley when $t=4 \mathrm{~s}$ if it starts from rest when $t=0 .$ Neglect the mass of the cord.

Key Concepts

Rotational Dynamics

This concept involves the study of objects in rotational motion and covers principles like torque, angular acceleration, and the relationship between them. In the context of this problem, it is crucial to understand how a net torque acting on a rotating body will cause angular acceleration, analogous to how net force causes linear acceleration in translational motion.

Moment of Inertia

The moment of inertia quantifies how the mass of a body is distributed relative to an axis of rotation, affecting the body's resistance to changes in its rotational state. For a thin disk, the moment of inertia is determined by a specific formula that links the mass of the disk to the square of its radius, which is essential in calculating the angular acceleration when a torque is applied.

Torque

Torque is the rotational equivalent of force and represents the tendency of a force to rotate an object about an axis. In this scenario, the different tensions applied on either side of the pulley create a net torque, which is calculated by multiplying the difference in these forces by the radius of the pulley.

Rotational Kinematics

This area of study deals with the equations that relate angular displacement, angular velocity, angular acceleration, and time. When the angular acceleration is constant, as it is when a constant net torque is applied, rotational kinematics can be used to determine the angular velocity of the object after a given time interval.

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