The 30 -lb flywheel A has a radius of gyration about its...

The 30 -lb flywheel $A$ has a radius of gyration about its center of 4 in. Disk $B$ weighs $50 \mathrm{lb}$ and is coupled to the flywheel by means of a belt which does not slip at its contacting surfaces. If a motor supplies a counterclockwise torque to the flywheel of $M=(50 t) \mathrm{lb} \cdot \mathrm{ft}$, where $t$ is in seconds, determine the time required for the disk to attain an angular velocity of $60 \mathrm{rad} / \mathrm{s}$ starting from rest.

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Key Concepts

Newton’s Second Law for Rotation

This principle states that the sum of torques on a rotating system is equal to the moment of inertia times the angular acceleration. It serves as the rotational analog to Newton’s second law for linear motion and is foundational for analyzing systems under applied torques.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to an axis of rotation and influences a body’s resistance to changes in its angular motion. Computing the total moment of inertia of a system is crucial for determining its response to applied torques.

Time-Dependent Torque and Integration

When a torque varies with time, the angular acceleration it produces is also time-dependent, requiring integration to relate the torque input over time to changes in angular velocity. This concept is key in solving differential equations that describe the evolution of rotational speed under nonconstant forcing.

Belt Coupling and Non-Slip Condition

In systems where components are connected by a belt that does not slip, the connected bodies share the same angular velocity. This non-slip condition ensures that the dynamics of each body are interdependent, meaning their moments of inertia contribute collectively to the overall rotational behavior.

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