Chapter Summary

13.1 Hooke’s Law: Stress and Strain Revisited

An oscillation is a back and forth motion of an object between two points of deformation.
An oscillation may create a wave, which is a disturbance that propagates from where it was created.
The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:
[latex]F = - \text{kx} ,[/latex]

where [latex]F[/latex] is the restoring force, [latex]x[/latex] is the displacement from equilibrium or deformation, and [latex]k[/latex] is the force constant of the system.
Elastic potential energy [latex]\text{PE}_{\text{el}}[/latex] stored in the deformation of a system that can be described by Hooke’s law is given by
[latex]\text{PE}_{\text{el}} = \left(\right. 1 / 2 \left.\right) \textit{kx}^{2} .[/latex]

Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
Maximum displacement is the amplitude [latex]X[/latex]. The period [latex]T[/latex] and frequency [latex]f[/latex] of a simple harmonic oscillator are given by[latex]T = 2π \sqrt{\frac{m}{k}}[/latex] and [latex]f = \frac{1}{2π} \sqrt{\frac{k}{m}}[/latex], where [latex]m[/latex] is the mass of the system.
Displacement in simple harmonic motion as a function of time is given by [latex]x \left(\right. t \left.\right) = X \text{cos} \frac{2π t}{T} .[/latex]
The velocity is given by[latex]v \left(\right. t \left.\right) = - v_{\text{max}} \text{sin} \frac{2π \text{ t}}{T}[/latex], where[latex]v_{\text{max}} = \sqrt{k / m} X[/latex].
The acceleration is found to be [latex]a \left(\right. t \left.\right) = - \frac{kX}{m} \text{ cos } \frac{2π t}{T} .[/latex]

A mass [latex]m[/latex] suspended by a wire of length [latex]L[/latex] is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about [latex]\text{15}º .[/latex]The period of a simple pendulum is
[latex]T = 2π \sqrt{\frac{L}{g}} ,[/latex]

where [latex]L[/latex] is the length of the string and [latex]g[/latex] is the acceleration due to gravity.

Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
[latex]\frac{1}{2} \text{mv}^{2} + \frac{1}{2} \text{kx}^{2} = \text{ constant}.[/latex]
Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses:
[latex]v_{\text{max}} = \sqrt{\frac{k}{m}} X .[/latex]

A projection of uniform circular motion undergoes simple harmonic oscillation.

Damped harmonic oscillators have non-conservative forces that dissipate their energy.
Critical damping returns the system to equilibrium as fast as possible without overshooting.
An underdamped system will oscillate through the equilibrium position.
An overdamped system moves more slowly toward equilibrium than one that is critically damped.

A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

A wave is a disturbance that moves from the point of creation with a wave velocity [latex]v_{\text{w}}[/latex].
A wave has a wavelength [latex]\lambda[/latex], which is the distance between adjacent identical parts of the wave.
Wave velocity and wavelength are related to the wave’s frequency and period by [latex]v_{\text{w}} = \frac{\lambda}{T}[/latex] or [latex]v_{\text{w}} = fλ .[/latex]
A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

Superposition is the combination of two waves at the same location.
Constructive interference occurs when two identical waves are superimposed in phase.
Destructive interference occurs when two identical waves are superimposed exactly out of phase.
A standing wave is one in which two waves superimpose to produce a wave that varies in amplitude but does not propagate.
Nodes are points of no motion in standing waves.
An antinode is the location of maximum amplitude of a standing wave.
Waves on a string are resonant standing waves with a fundamental frequency and can occur at higher multiples of the fundamental, called overtones or harmonics.
Beats occur when waves of similar frequencies [latex]f_{1}[/latex] and [latex]f_{2}[/latex] are superimposed. The resulting amplitude oscillates with a beat frequency given by [latex]f_{\text{B}} = \mid f_{1} - f_{2} \mid .[/latex]

Intensity is defined to be the power per unit area: [latex]I = \frac{P}{A}[/latex] and has units of [latex]\text{W}/\text{m}^{2}[/latex].

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