Sections

Section Summary
# Section Summary

### 16.1 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:
$$F=-\text{kx},$$
where $F$ is the restoring force, $x$ is the displacement from equilibrium or deformation, and $k$ is the force constant of the system.

- Elastic potential energy ${\text{PE}}_{\text{el}}$ stored in the deformation of a system that can be described by Hooke’s law is given by
$${\text{PE}}_{\text{el}}=(1/2){\mathit{\text{kx}}}^{2}.$$

### 16.2 Period and Frequency in Oscillations

- Periodic motion is a repetitious oscillation.
- The time for one oscillation is the period $\mathrm{T.}$
- The number of oscillations per unit time is the frequency
*$\mathrm{f.}$* - These quantities are related by
$$f=\frac{1}{T}.$$

### 16.3 Simple Harmonic Motion: A Special Periodic Motion

- 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
*$X$*. The period*$T$*and frequency $f$ of a simple harmonic oscillator are given by$T=\mathrm{2\pi}\sqrt{\frac{m}{k}}$ and $f=\frac{1}{\mathrm{2\pi}}\sqrt{\frac{k}{m}}$, where $m$ is the mass of the system.

- Displacement in simple harmonic motion as a function of time is given by $x(t)=X\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\frac{\mathrm{2\pi}t}{T}.$
- The velocity is given by $v(t)=-{v}_{\text{max}}\text{sin}\frac{\mathrm{2\pi}\text{t}}{T}$, where ${v}_{\text{max}}=\sqrt{k/m}X$.
- The acceleration is found to be $a(t)=-\frac{\mathrm{kX}}{m}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\frac{\mathrm{2\pi}t}{T}.$

### 16.4 The Simple Pendulum

- A mass
*$m$*suspended by a wire of length $L$ is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about $\text{15\xba}.$The period of a simple pendulum is

$$T=\mathrm{2\pi}\sqrt{\frac{L}{g}},$$where $L$ is the length of the string and $g$ is the acceleration due to gravity.

### 16.5 Energy and the Simple Harmonic Oscillator

- Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
$$\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\text{kx}}^{2}=\text{constant.}$$
- 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.
$${v}_{\text{max}}=\sqrt{\frac{k}{m}}X$$

### 16.6 Uniform Circular Motion and Simple Harmonic Motion

A projection of uniform circular motion undergoes simple harmonic oscillation.

### 16.7 Damped Harmonic Motion

- 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.

### 16.8 Forced Oscillations and Resonance

- 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.

### 16.9 Waves

- A wave is a disturbance that moves from the point of creation with a wave velocity ${v}_{\text{w}}$.
- A wave has a wavelength $\lambda $, 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 ${v}_{\text{w}}=\frac{\lambda}{T}$ or ${v}_{\text{w}}=\mathrm{f\lambda}.$
- 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.

### 16.10 Superposition and Interference

- 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 ${f}_{1}$ and ${f}_{2}$ are superimposed. The resulting amplitude oscillates with a beat frequency given by
$${f}_{\text{B}}=\mid {f}_{1}-{f}_{2}\mid .$$

### 16.11 Energy in Waves: Intensity

Intensity is defined to be the power per unit area:

$I=\frac{P}{A}$ and has units of ${\text{W/m}}^{2}$.