Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
College Physics for AP® Courses

Section Summary

College Physics for AP® CoursesSection 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=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} } } {} .

16.2 Period 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} } } {}

16.3 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 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=mkT=mk size 12{T=2π sqrt { { {m} over {k} } } } {} and f=1kmf=1km, 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 t T . x ( t ) = X cos t T . size 12{x \( t \) =X"cos" { {2π`t} over {T} } } {}
  • The velocity is given by v(t)= vmax sin tTv(t)= vmax sin tT, where vmax =k/mX vmax =k/mX.
  • The acceleration is found to be a ( t ) = kX m cos t T . a ( t ) = kX m cos t T . size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}

16.4 The 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=Lg,T=Lg, 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.

16.5 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:
    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} {}

16.6 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 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 = . v w = . 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.

16.10 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 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 } {}

16.11 Energy 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} } } {}.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/college-physics-ap-courses/pages/1-connection-for-ap-r-courses
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/college-physics-ap-courses/pages/1-connection-for-ap-r-courses
Citation information

© Mar 3, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.