OSCILLATIONS

 

  1. Definitions
    1. Any motion that repeats itself in equal periods of time - Periodic Motion
    2. If the motion can be described in terms of sines and cosines - Harmonic Motion
    3. Motion that is back and forth over the same path - Oscillatory
    4. Period (T) - time for a round trip
    5. Frequency: f = 1/T
    6. Equilibrium position - no net force
    7. Displacement is measured with respect to the equilibrium position
  1. One-dimensional motion
    1. General ideas
    1. At equilibrium, x = 0, and there is no unbalanced force, F = 0
    2. If the forces are conservative, the total mechanical energy is constant

 

E = KE + PE = constant

 

    1. For a mass attached to a spring


 

 

F = -kx

k = spring constant (Hooke's Law)

    1. This simple harmonic oscillator (SHO) undergoes simple harmonic motion (SHM)
    2. We can associate a potential energy (PE) with a SHO
    3. F = -kx = -(dU/dx)

      dU = kxdx

      PE =U

      U = (1/2)kx2

      (after an integration)

    4. Since,
    5. FNET = ma = m(dx2/dt2)= -kx

      m(dx2/dt2) + kx = 0

    6. Solution,
    7. x = Acos(w t + d )

      where A, w , and d are constants

    8. Substituting this into the original differential equation we find w 2 = k/m
    9. Motion repeats itself after a time of 2p /w and
    10. T = 2p /w = 1/f

      f = (1/2p )(k/m)(1/2)

    11. Energy considerations

x = Acos(w t + d )

U = (1/2)kx2 = (1/2)k[Acos(w t + d )]2

KE = (1/2)mv2 = (1/2)m(dx/dt)2

KE = (1/2)mw 2 [Asin(w t + d )]2

E = KE + PE

E = (1/2)kA2

(How did we get this?)

Also,

v = dx/dt = ± [(k/m)(A2-x2)](1/2)