# Simple Harmonic Motion Calculator

Created by Bogna Szyk
Reviewed by Rijk de Wet
Last updated: Jan 12, 2023

This simple harmonic motion calculator will help you find the displacement, velocity, and acceleration of an oscillating particle. All you need to do is determine the fundamental properties of the periodic motion (for example, its frequency and amplitude) and input them into the simple harmonic motion equations.

If you're interested in how a harmonic wave propagates in space, check out the harmonic wave equation calculator instead.

## How to use the simple harmonic motion calculator

The simple harmonic motion calculator is easy to use. Follow these simple steps:

1. Enter the parameters of the harmonic motion that you know. These would typically be:

• The motion's amplitude ($A$);

• The motion's frequency ($f$); and

• The time at which you want to capture the motion's other values ($t$).

2. The simple harmonic motion calculator will calculate the values describing the motion at time $t$ for you. These are:

• The angular frequency ($\omega$);

• The displacement ($y$);

• The velocity ($v$); and

• The acceleration ($a$).

## What is the simple harmonic motion?

Simple harmonic motion is a type of oscillating motion. A particle in this kind of motion always moves along the same path, with an acceleration towards a fixed point.

A great example of such a case can be the simple pendulum. A mass suspended on a string that is moved from its equilibrium position will move back and forth, always accelerating toward the equilibrium point. If we assume no energy loss, the pendulum will keep swinging forever. At the equilibrium point, the kinetic energy is maximal (see kinetic energy calculator).

## Simple harmonic motion equations

If you know the period of oscillations, it is possible to calculate the position, velocity, and acceleration of the particle at every single point in time. All you have to do is to apply the following simple harmonic motion equations:

\small \begin{aligned} {\scriptsize\bullet}\quad y &= A \cdot \sin(\omega t) \\ {\scriptsize\bullet}\quad v &= A \cdot \omega \cdot \cos(\omega t) \\ {\scriptsize\bullet}\quad a &= -A \cdot \omega^2 \cdot \sin(\omega t) \end{aligned}

where:

• $A$ is the amplitude of oscillations;
• $\omega$ is the angular frequency of oscillations in $\text{rad}/\text{s}$;
• $t$ is the time at which you measure the particle's displacement;
• $y$ is the displacement;
• $v$ is the velocity; and
• $a$ is the acceleration.

$\omega$ can be calculated as $\omega = 2\pi f$, where $f$ is the frequency.

Our simple harmonic motion calculator uses these equations to work out the values other than what you give it.

## How to calculate the acceleration of an oscillating particle?

Let's analyze the example of a particle in simple harmonic motion. We want to find out what is the displacement, velocity, and acceleration at $t = 1.4 \text{ s}$.

1. Determine the frequency of oscillations. Let's assume it's equal to $f = 1 \text{ Hz}$.

2. Calculate the angular frequency using the formula for angular frequency:

$\omega = 2\pi f = 6.28 \text{ rad/s}$.

3. Decide on what is the amplitude of the oscillations. Let's take $A = 15 \text{ mm}$.

4. Plug all of the values into the simple harmonic motion equations:

\footnotesize\qquad \begin{aligned} y &= 15 \cdot \sin(6.28 \cdot 1.4) \\ &= 8.82 \text{ mm} \\[1em] v &= 15 \cdot 6.28 \cdot \cos(6.28 \cdot 1.4) \\ &= -76.25 \text{ mm/s} \\[1em] a &= - 15 \cdot (6.28)^2 \cdot \sin(6.28 \cdot 1.4) \\ &= -348 \text{mm/s}^2 \\ \end{aligned}

Of course, instead of calculating it manually, you can plug all of the values into the simple harmonic motion calculator.

Bogna Szyk
Amplitude (A)
in
Time (t)
sec
Frequency (f)
Hz
Angular frequency (ω)
per s
Displacement (y)
in
Velocity (v)
m/s
Acceleration (a)
m/s²
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