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15.1 Simple Harmonic Motion

  • Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by f=1Tf=1T.
  • Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
  • Maximum displacement is the amplitude A. The angular frequency ωω, period T, and frequency f of a simple harmonic oscillator are given by ω=kmω=km, T=2πmk,andf=12πkmT=2πmk,andf=12πkm, where m is the mass of the system and k is the force constant.
  • Displacement as a function of time in SHM is given byx(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ)x(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ).
  • The velocity is given by v(t)=Aωsin(ωt+ϕ)=vmaxsin(ωt+ϕ),v(t)=Aωsin(ωt+ϕ)=vmaxsin(ωt+ϕ), where vmax=Aω=Akmvmax=Aω=Akm.
  • The acceleration is a(t)=Aω2cos(ωt+ϕ)=amaxcos(ωt+ϕ)a(t)=Aω2cos(ωt+ϕ)=amaxcos(ωt+ϕ), where amax=Aω2=Akmamax=Aω2=Akm.

15.2 Energy in Simple Harmonic Motion

  • The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
  • Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given byU=12kx2.U=12kx2.
  • Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
  • The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using

15.3 Comparing Simple Harmonic Motion and Circular Motion

  • A projection of uniform circular motion undergoes simple harmonic oscillation.
  • Consider a circle with a radius A, moving at a constant angular speed ωω. A point on the edge of the circle moves at a constant tangential speed of vmax=Aωvmax=Aω. The projection of the radius onto the x-axis is x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ), where (ϕ)(ϕ) is the phase shift. The x-component of the tangential velocity is v(t)=Aωsin(ωt+ϕ)v(t)=Aωsin(ωt+ϕ).

15.4 Pendulums

  • A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about 15°15°. The period of a simple pendulum is T=2πLgT=2πLg, where L is the length of the string and g is the acceleration due to gravity.
  • The period of a physical pendulum T=2πImgLT=2πImgL can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
  • The period of a torsional pendulum T=2πIκT=2πIκ can be found if the moment of inertia and torsion constant are known.

15.5 Damped Oscillations

  • 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.

15.6 Forced Oscillations

  • A system’s natural frequency is the frequency at which the system oscillates 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.
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