Simple Harmonic Motion

# Syntax # Overview

Simple Harmonic Motion describes when a particle or object is in a constant motion that is repetitive in its path. It will oscillate back and forth on a particular path constantly, and theoretically forever. One property of the oscillation is its frequency, which is how many oscillations that are completed per second. There is also the period, T, which is time for one complete oscillation.

# Equations

## Equation One

x(t) = Acos(wt)
A = amplitude
w = angular frequency

## Equation Two

v(t) = Awsin(wt)

++Equation Three
a(t) = -Aw^2cos(wt)

T = 1/f

## Equation Five

w = 2(pi)/T = 2(pi)f

A 20 g particle moves in simple harmonic motion with a frequency of 3 oscillations per second and an amplitude of 5cm.
(a) Through what total distance does the particle move during one cycle of its motion?

A 20 g particle moves in simple harmonic motion with a frequency of 3 oscillations per second and an amplitude of 5cm.
(b) What is its maximum speed? Where does that occur?

# Special Cases

A special case of simple harmonic motion is in the case of pendulums. Pendulums operate in the same manner, but include gravity in their calculations. Also, it is important to note that while theoretically, in a frictionless system something may oscillate forever, most systems do eventually come to rest due to friction.

# Applications The most obvious application is when looking a systems involving springs that have been given energy and released. These equations help you to calculate the positions, velocity, and acceleration of objects attached to the spring.