Simple Harmonic Oscillator

Summary

• Oscillatory motion is repetitive and fluctuates between two locations.
  • For one dimensional motion, the maximum displacement of an oscillatory system from its equilibrium position is called its amplitude.
    • The symbol for amplitude is A (italic capital a).
    • The SI unit of amplitude is the meter [m], but other length units may be used.
• Periodic motion repeats a cycle of motion with a characteristic time.
  • A period is the time to complete one cycle of periodic motion.
    • The symbol for period is T (italic capital t).
    • The SI unit of period is the second [s].
    • Time (t), number of cycles (n), and period (T) are related by the following equation…

      T =	t
      	n

  • Frequency is the number of cycles that elapse in a unit of time.
  • The symbol for frequency is f (long f or italic lowercase f).
  • The SI unit of frequency is the hertz [Hz], which is equal to an inverse second.

    ⎡ ⎢ ⎣	Hz =	1		= s−1	⎤ ⎥ ⎦
    		s

  • Number (n), time (t), and frequency (f) are related by the following equation…

    f =	n
    	t

  • Frequency and period are inverse quantities.

    f =	1	⇔	T =	1
    	T			f

  • Phase, phase angle, or phase shift is the state of progresssion of a periodic system.
    • The symbol for phase is φ (lowercase phi).
    • The SI unit of phase is the radian [rad], but degrees may also be used (°).
    • Phase can also be decribed as a fraction of a cycle or a period.
    • One complete cycle corresponds to 2π radian.
  • Angular frequency is the rate of change of phase with time.
    • The symbol for angular frequency is ω (lowercase omega).
    • The SI unit of angular freqeuncy is the radian per second [rad/s].
    • Angular freqeuncy converts time into radians for use in equations containing sine and cosine.
    • Angular frequency and frequency are related by the following equation…

      ω = 2πf

  • Oscillatory motion that is not periodic is said to be aperiodic.
• A simple harmonic oscillator (abbreviated sho) is any mechanical system in which the net force on the system…
  • is directly proportional to the displacement of the system from its equilibrium position
  • is a restoring force (acts in a direction opposite the displacement)

    ∑F = −kx

• Simple harmonic motion (abbreviated shm)…
  • is what a simple harmonic oscillator does when it is disturbed from its equilibrium position.
• is both oscillatory and periodic.
• is described by the following equation…

  x = A sin(2πft + φ)

  where…

  x =	position [m, cm, etc.]
  A =	amplitude [m, cm, etc.]
  f =	frequency [Hz]
  t =	time [s]
  φ =	phase [rad]