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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:

    where FF is the restoring force, xx is the displacement from equilibrium or deformation, and kk is the force constant of the system.

  • Elastic potential energy PEelPEel 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 .

16.2 Period and Frequency in Oscillations

  • Periodic motion is a repetitious oscillation.
  • The time for one oscillation is the period TT.
  • The number of oscillations per unit time is the frequency ff.
  • These quantities are related by
    f = 1 T . f = 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 XX. The period TT and frequency ff of a simple harmonic oscillator are given by

    T=mkT=mk and f=1kmf=1km, where mm 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 .
  • 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 .

16.4 The Simple Pendulum

  • A mass mm suspended by a wire of length LL is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about 15º.15º.

    The period of a simple pendulum is


    where LL 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.
  • 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 .

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.
  • A wave has a wavelength λλ, 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 or v w = . v w = .
  • 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 and f2f2 are superimposed. The resulting amplitude oscillates with a beat frequency given by
    f B = f 1 f 2 . f B = f 1 f 2 .

16.11 Energy in Waves: Intensity

Intensity is defined to be the power per unit area:

I=PAI=PA and has units of W/m2W/m2.

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