The 50 -lb wheel has a radius of gyration about its center...

The 50 -lb wheel has a radius of gyration about its center of gravity $G$ of $k_{G}=0.7 \mathrm{ft}$. If it rolls without slipping. determine its angular velocity when it has rotated clockwise $90^{\circ}$ from the position shown. The spring $A B$ has a stiffness $k=1.20 \mathrm{lb} / \mathrm{ft}$ and an unstretched length of $0.5 \mathrm{ft}$. The wheel is released from rest.

Key Concepts

Rolling Without Slipping

This concept describes a condition where the point of contact between a rolling object and the surface is momentarily at rest relative to the surface. It links the translational motion of the object’s center of mass with its rotational motion, establishing a geometric relationship between linear displacement and angular displacement, which is essential when analyzing the motion when no slipping occurs.

Moment of Inertia and Radius of Gyration

The moment of inertia quantifies an object’s resistance to changes in its rotational motion about an axis. The radius of gyration is a related concept that provides a measure of how the mass of an object is distributed with respect to the axis of rotation. It simplifies the calculation of the moment of inertia for complex bodies and is crucial in problems dealing with rotational dynamics.

Angular Velocity and Angular Displacement

Angular velocity measures the rate of change of the angular displacement of an object, representing how quickly an object rotates. Angular displacement quantifies the angle through which a point, line, or body is rotated in a specified sense about a given axis. These concepts are central to understanding the rotational dynamics of any system.

Energy Conservation in Rotational Dynamics

The principle of energy conservation states that the total mechanical energy in an isolated system remains constant unless external work is done. In rotational dynamics, this principle is applied by equating the changes in potential energy (including gravitational and elastic potential energy) to the rotational kinetic energy, enabling the determination of key parameters like angular velocity in systems undergoing rolling motion.

Spring Potential Energy

Spring potential energy arises when a spring is either compressed or extended from its natural length. It is quantified by the stiffness (or spring constant) and the displacement from equilibrium. In problems involving elastic elements like springs, this energy plays a critical role in the overall energy balance and influences the subsequent motion of the mechanical system.

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