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# 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]