# Definition:Simple Harmonic Motion

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## Contents

## Definition

Consider a physical system $S$ whose motion can be expressed in the form of the following equation:

- $x = A \sin \left({\omega t + \phi}\right)$

where $A$ and $\phi$ are constants.

Then $S$ is in a state of **simple harmonic motion**.

### Amplitude

The parameter $A$ is known as the **amplitude** of the motion.

### Period

The **period** $T$ of the motion of $S$ is the time required for one complete cycle:

- $T = \dfrac {2 \pi} \omega$

### Frequency

The **frequency** $\nu$ of the motion of $S$ is the number of complete cycles per unit time:

- $\nu = \dfrac 1 T = \dfrac \omega {2 \pi}$

## Also defined as

**Simple harmonic motion** can also be characterised in the form:

- $x = A \cos \left({\omega t + \phi}\right)$

From Sine of Angle plus Right Angle:

- $\sin \left({\omega t + \phi + \dfrac \pi 2}\right) = \cos \left({\omega t + \phi}\right)$

the two forms can be seen to be equivalent.

## Also known as

**Simple harmonic motion** can also be referred to as **simple harmonic oscillation** or **simple harmonic vibration**.

Some sources abbreviate it to **SHM**.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems