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=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
    PEel=(1/2)kx2.PEel=(1/2)kx2 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 T.T. size 12{T} {}
  • The number of oscillations per unit time is the frequency f.f. size 12{f} {}
  • These quantities are related by
    f=1T.f=1T. 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)=XcostT.x(t)=XcostT. size 12{x \( t \) =X"cos" { {2π`t} over {T} } } {}
  • The velocity is given by v(t)=vmaxsin tTv(t)=vmaxsin tT, where vmax=k/mXvmax=k/mX.
  • The acceleration is found to be a(t)=kXm cos tT.a(t)=kXm cos tT. 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:
    12mv2+12kx2= constant.12mv2+12kx2= 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.
    vmax=kmXvmax=kmX 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 vw=λTvw=λT size 12{v size 8{w}= { {λ} over {T} } } {} or vw=.vw=. 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
    fB=f1f2.fB=f1f2. 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} } } {}.