A 4-kg disk A is mounted on arm B C, which has a negligible...

A $4-\mathrm{kg}$ disk $A$ is mounted on arm $B C$, which has a negligible mass. If a torque of $M=\left(5 e^{0.5 t}\right) \mathrm{N} \cdot \mathrm{m}$, where $t$ is in seconds, is applied to the arm at $C$, determine the angular velocity of $B C$ in 2 s starting from rest. Solve the problem assuming that (a) the disk is set in a smooth bearing at $B$ so that it moves with curvilinear translation, (b) the disk is fixed to the shaft $B C$, and (c) the disk is given an initial freely spinning angular velocity of $\omega_{D}=\{-80 \mathbf{k}] \mathrm{rad} / \mathrm{s}$ prior to application of the torque.

Key Concepts

Initial Angular Velocity

An initial angular velocity influences the subsequent motion when an external torque is applied. Including scenarios where parts of the system already have rotational motion (such as a disk spinning before the application of torque) is important because it alters the net angular momentum and modifies the resulting angular velocities after additional forces are introduced.

System Constraints and Mounting Conditions

The way in which a rotating body is mounted or connected to other parts of a system can significantly alter its response to an applied torque. Whether the disk is free to rotate independently in a bearing, fixed directly to the arm, or exhibits a pre-existing spin, these constraints affect the coupling between translational and rotational motion and the overall dynamic behavior of the system.

Rotational Dynamics

This concept involves the relationship between applied torque, angular acceleration, and the rotational inertia of a body. In problems involving rotation such as this one, knowing that torque equals moment of inertia times angular acceleration allows for the determination of angular velocity when torque is applied over time.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a given axis. For a disk or any rotating body, its value depends on how the mass is distributed relative to the axis of rotation, which is essential when calculating the effects of an applied torque.

Integration of Time?Varying Torque

When torque is a function of time instead of being constant, determining the angular velocity requires integrating the angular acceleration over time. This process involves dividing the time-dependent torque by the moment of inertia to find the instantaneous angular acceleration and then integrating that with respect to time.

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