College Physics for AP® Courses

# Section Summary

### 16.1Hooke’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:
$F=−kx,F=−kx, size 12{F= - ital "kx"} {}$

where $FF size 12{F} {}$ is the restoring force, $xx size 12{x} {}$ is the displacement from equilibrium or deformation, and $kk size 12{k} {}$ is the force constant of the system.

• Elastic potential energy $PEelPEel size 12{"PE" rSub { size 8{"el"} } } {}$ stored in the deformation of a system that can be described by Hooke’s law is given by
$PE el = ( 1 / 2 ) kx 2 . PE el = ( 1 / 2 ) kx 2 size 12{ ital "PE" rSub { size 8{e1} } = $$1/2$$ ital "kx" rSup { size 8{2} } } {} .$

### 16.2Period and Frequency in Oscillations

• Periodic motion is a repetitious oscillation.
• The time for one oscillation is the period $TT size 12{T} {}$.
• The number of oscillations per unit time is the frequency $ff size 12{f} {}$.
• These quantities are related by
$f = 1 T . f = 1 T . size 12{f= { {1} over {T} } } {}$

### 16.3Simple 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 $XX size 12{X} {}$. The period $TT size 12{T} {}$ and frequency $ff size 12{f} {}$ of a simple harmonic oscillator are given by

$T=2πmkT=2πmk size 12{T=2π sqrt { { {m} over {k} } } } {}$ and $f=12πkmf=12πkm$, where $mm size 12{m} {}$ is the mass of the system.

• Displacement in simple harmonic motion as a function of time is given by $x ( t ) = X cos 2π t T . x ( t ) = X cos 2π t T . size 12{x $$t$$ =X"cos" { {2π`t} over {T} } } {}$
• The velocity is given by $v(t)=− vmax sin2π tTv(t)=− vmax sin2π tT$, where $vmax =k/mX vmax =k/mX$.
• The acceleration is found to be $a ( t ) = − kX m cos 2π t T . a ( t ) = − kX m cos 2π t T . size 12{a $$t$$ = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}$

### 16.4The Simple Pendulum

• A mass $mm size 12{m} {}$ suspended by a wire of length $LL size 12{L} {}$ is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about $15º.15º size 12{"15"°} {}.$

The period of a simple pendulum is

$T=2πLg,T=2πLg, size 12{T=2π sqrt { { {L} over {g} } } } {}$

where $LL size 12{L} {}$ is the length of the string and $gg$ is the acceleration due to gravity.

### 16.5Energy 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:
$1 2 mv 2 + 1 2 kx 2 = constant. 1 2 mv 2 + 1 2 kx 2 = constant. size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } ital "kx" rSup { size 8{2} } =" constant"} {}$
• 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:
$v max = k m X . v max = k m X . size 12{v rSub { size 8{"max"} } = sqrt { { {k} over {m} } } X} {}$

### 16.6Uniform Circular Motion and Simple Harmonic Motion

A projection of uniform circular motion undergoes simple harmonic oscillation.

### 16.7Damped 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.

### 16.8Forced 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.

### 16.9Waves

• A wave is a disturbance that moves from the point of creation with a wave velocity $vwvw size 12{v rSub { size 8{w} } } {}$.
• A wave has a wavelength $λλ size 12{λ} {}$, 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 $v w = λ T v w = λ T size 12{v size 8{w}= { {λ} over {T} } } {}$ or $v w = fλ . v w = fλ . size 12{v size 8{w}=fλ} {}$
• 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.

### 16.10Superposition 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 $f1f1 size 12{f rSub { size 8{1} } } {}$ and $f2f2 size 12{f rSub { size 8{2} } } {}$ are superimposed. The resulting amplitude oscillates with a beat frequency given by
$f B = ∣ f 1 − f 2 ∣ . f B = ∣ f 1 − f 2 ∣ . size 12{f rSub { size 8{B} } = lline f rSub { size 8{1} } - f rSub { size 8{2} } rline } {}$

### 16.11Energy in Waves: Intensity

Intensity is defined to be the power per unit area:

$I=PAI=PA size 12{I= { {P} over {A} } } {}$ and has units of $W/m2W/m2 size 12{"W/m" rSup { size 8{2} } } {}$.

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