47. Wheel Rotates A wheel is free to rotate about its fixed...

47. Wheel Rotates A wheel is free to rotate about its fixed axle. A spring is attached to one of its spokes a distance $r$ from the axle, as shown in Fig. $16-42$ (a) Assuming that the wheel is a hoop of mass $m$ and radius $R$, obtain the angular frequency of small oscillations of this system in terms of $m, R, r$, and the spring constant $k$. How does the result change if (b) $r=R$ and (c) $r=0$ ?

47. Wheel Rotates A wheel is free to rotate about its fixed axle. A spring is attached to one of its spokes a distance $r$ from the axle, as shown in Fig. $16-42$
(a) Assuming that the wheel is a hoop of mass $m$ and radius $R$, obtain the angular frequency of small oscillations of this system in terms of $m, R, r$, and the spring constant $k$. How does the result change if (b) $r=R$ and (c) $r=0$ ?

Key Concepts

Moment of Inertia

Moment of inertia quantifies an object’s resistance to angular acceleration around a given axis. For a hoop with mass m and radius R, the moment of inertia is given by I = mR². This parameter is critical in analyzing rotational systems because it plays the role analogous to mass in linear motion, determining how much torque is needed to produce a desired angular acceleration.

Restoring Torque from a Spring

When a spring is attached to a rotating spoke at a distance r from the axis, any angular displacement leads to a linear displacement at the point of attachment, causing the spring to exert a force. This force, acting at a distance, produces a torque. The magnitude of this restoring torque, derived from the product of the spring force and the lever arm (r), is central to setting up the rotational analog of Hooke’s law in this system.

Angular Frequency of Small Oscillations

For small angular displacements, the system exhibits simple harmonic motion. The angular frequency of these oscillations is determined by the ratio of the effective restoring torque (which depends on the spring constant and the lever arm squared) to the moment of inertia. This leads to an expression where the angular frequency is proportional to the square root of the effective spring constant divided by the moment of inertia, providing insights into the dynamics and stability of the oscillating system.

Effect of Lever Arm Length

The distance r at which the spring is attached to the wheel has a direct impact on the system’s behavior. When r equals R, the maximum lever arm is used, yielding a particular angular frequency expression. Conversely, when r is zero, the spring is attached at the axis of rotation, which means no torque is generated regardless of the spring’s state of extension. This variation illustrates the importance of the point of application of the force in determining the dynamics of rotational systems.

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