A ball of mass m is connected to two rubber bands of length...

A ball of mass $m$ is connected to two rubber bands of length $L$, each under tension $T$ as shown in Figure $\mathrm{P} 15.39 .$ The ball is displaced by a small distance $y$ perpendicular to the length of the rubber bands. Assuming the tension does not change, show that (a) the restoring force is $-(2 T / L) y$ and (b) the system exhibits simple harmonic motion with an angular frequency $\omega=\sqrt{2 T / m L}$

Key Concepts

Restoring Force and Hooke’s Law

In many mechanical systems, the net force acting to return an object to its equilibrium position is known as the restoring force. When this force is directly proportional to the displacement from equilibrium and directed opposite to it, the behavior of the system is analogous to Hooke’s law, where F = -kx. In this problem, by projecting the constant tension forces along the direction of displacement, one obtains a net force proportional to the displacement, thereby reproducing a restoring force of the form –(2T/L)y.

Small Displacement Approximation

When an object is displaced by a small amount from its equilibrium, the nonlinearities in the equations governing the system’s behavior can often be neglected. This linearization, known as the small displacement approximation, simplifies the analysis and allows the use of first-order approximations for trigonometric functions. In the given example, this approximation makes it possible to assume that the tension’s direction changes only slightly, leading to a linear relationship between the restoring force and the displacement.

Simple Harmonic Motion and Angular Frequency

A system exhibits simple harmonic motion (SHM) when the restoring force is directly proportional to the displacement and directed towards the equilibrium position, resulting in oscillatory motion. The angular frequency of this motion is determined by the ratio of the effective force constant to the mass of the oscillating object. In the provided problem, the derived net restoring force of –(2T/L)y leads to a SHM with an angular frequency ? = ?(2T/(mL)), which characterizes the rate of oscillation for the system.

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