# 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.

- The symbol for amplitude is

- For one dimensional motion, the maximum displacement of an oscillatory system from its equilibrium position is called its amplitude.
- 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*

- The symbol for period is
- 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 = 2π

*f*

- Oscillatory motion that is not periodic is said to be aperiodic.

- A period is the time to complete one cycle of periodic motion.
- 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…

where…*x*=*A*sin(2π*ft*+ φ)*x*=position [m, cm, etc.] *A*=amplitude [m, cm, etc.] *f*=frequency [Hz] *t*=time [s] φ = phase [rad]