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Chapter 10 Learning Objectives

Concepts and Skills to Review

  • Hooke's law (Section 2.1)
  • Graphical relationship of position, velocity, and acceleration (Sections 3.2, 3.3)
  • Elastic potential energy (Section 6.5)

Summary

  • A deformation is a change in the size or shape of an object.
  • When deforming forces are removed, an elastic object returns to its original shape and size.
  • Hooke's law, in a generalized form, says that the deformation of a material (measured by the strain) is proportional to the magnitude of the forces causing the deformation (measured by the stress). The definitions of stress and strain are as given in the table:
     Type of deformation
     Tensile or CompressiveShearVolume
    StressForce per unit cross-sectional area F/AShear force divided by the parallel area of the surface on which it acts F/APressure P
    Strainfractional length change ΔL/Lratio of the relative displacement Δx to the separation L of the two parallel surfaces Δx/Lfractional volume change ΔV/V
    Constant of proportionalityYoung's modulus YShear modulus SBulk modulus B
  • If the tensile or compressive stress exceeds the proportional limit, the strain is no longer proportional to the stress. The solid still returns to its original length when the stress is removed as long as the stress does not exceed the elastic limit. If the stress exceeds the elastic limit, the material is permanently deformed. For larger stresses yet, the solid fractures when the stress reaches the breaking point. The maximum stress that can be withstood without breaking is called the ultimate strength.
  • Vibrations occur in the vicinity of a point of stable equilibrium. An equilibrium point is stable if the net force on an object when it is displaced from equilibrium points back toward the equilibrium point. Such a force is called a restoring force since it tends to restore equilibrium.
  • Simple harmonic motion is periodic motion that occurs whenever the restoring force is proportional to the displacement from equilibrium. In SHM, the position, velocity, and acceleration as functions of time are sinusoidal (i.e., sine or cosine functions). Any oscillatory motion is approximately SHM if the amplitude is small, because for small oscillations the restoring force is approximately linear.
  • The maximum velocity and acceleration in SHM are
      vm = ωA and am = ω2A(10-21,10-22)
    where ω is the angular frequency. The acceleration is proportional to and in the opposite direction from the displacement:
      ax(t) = -ω2x(t)(10-19)
  • The equations of motion for SHM are:
    If x = A at t = 0,
    x = A cos ωt
    vx = -vm sin ωt
    ax = -am cos ωt
    If x = 0 at t = 0,
    x = A sin ωt
    vx = vm cos ωt
    ax = -am sin ωt
    In either case, the velocity is 1/4 of a cycle ahead of the position and the acceleration is 1/4 of a cycle ahead of the velocity.
  • The angular frequency for a mass-spring system is
     <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57993/image10_20a.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> (10-20a)
    For a simple pendulum it is
     <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57993/image10_26a.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> (10-26a)
    and for physical pendulum it is
     <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57993/image10_28a.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> (10-28a)
  • In the absence of dissipative forces, the total mechanical energy of a simple harmonic oscillator is constant and proportional to the square of the amplitude:
     <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57993/image10_13.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a> (10-13)
    where the potential energy has been chosen to be zero at the equilibrium point. At any point, the sum of the kinetic and potential energies is constant:
     <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57993/image10_12.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> (10-12)







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