# Simple Harmonic Motion Calculator

Created by Krishna Nelaturu
Last updated: Sep 05, 2022

Our simple harmonic motion calculator can help you determine the displacement, velocity, and acceleration of an oscillating body at any given instant. Join us in a brief discussion below on the simple harmonic motion as we look at:

• Simple harmonic motion definition.
• Simple harmonic motion formulae.
• An example calculation.

There are many similarities between simple harmonic motion equations and uniform circular motion. We encourage you to go through our circular motion calculator before going any further.

## Let's define simple harmonic motion

Consider a small mass attached to a spring horizontally. At equilibrium, the mass is at rest. If we stretch the mass horizontally, it will start oscillating horizontally as the spring attempts to restore it to equilibrium (see: Hooke's law). This oscillation is an example of a simple harmonic motion.

Simple harmonic motion refers to an object's oscillation about an equilibrium state due to a restoring force.

A simple pendulum is another example of simple harmonic motion.

## Simple harmonic motion equations

The simple harmonic equations relate displacement, velocity, and acceleration to amplitude, angular frequency, and time. The displacement is given by:

$y = A \cdot \sin(\omega t)$

Where:

• $y$ - Displacement from the equilibrium position;
• $A$ - Amplitude of oscillation (maximum displacement);
• $\omega$ - Angular frequency of the oscillation;
• $t$ - Time instant at which we want to measure the displacement.

The angular frequency $(\omega)$ is related to the frequency $(f)$ by:

$\omega = 2 \pi f$

We can obtain a velocity equation for simple harmonic motion by differentiating the displacement equation with respect to time:

$v = \frac{d}{dt}\left(A \cdot \sin(\omega t)\right) \\[1em] v = A \cdot \omega \cdot \cos(\omega t)\\$

Where $v$ is the velocity of the oscillating particle.

Likewise, we get the acceleration by differentiating velocity with respect to time:

$a = \frac{d}{dt}\left(A \cdot \omega \cdot \cos(\omega t)\right) \\[1em] a = -A \cdot \omega^2 \cdot\sin(\omega t)$

Where $a$ is the acceleration of the oscillating particle.

## How to calculate simple harmonic motion parameters

Let's apply these simple harmonic formulae to an example problem. If a body oscillates with an amplitude of $3 \text{ cm}$ at a frequency of $2.9 \text{ Hz}$, what are its displacement, velocity, and acceleration $4 \text{ seconds}$ into the oscillation?

Given data:

• Amplitude $A = 3 \text{ cm}$.
• Frequency $f = 2.9 \text{ Hz}$.
• Time $t = 4 \text{ s}$.

To find:
Displacement, velocity, and acceleration

Let's start by finding the angular frequency:

$\omega = 2 \pi f = 2 \pi \cdot (2.9)\\ \omega = 18.22 \text{ rad/s}$

Using the displacement formula for simple harmonic motion:

\begin{align*} y &= A \cdot \sin(\omega t)\\ &= 3 \cdot \sin(18.22 \cdot 4)\\ y&= -1.76 \text{ cm} \end{align*}

Using the velocity equation for simple harmonic motion:

\begin{align*} v &= A \cdot \omega \cdot \cos(\omega t)\\ &= 3 \cdot 18.22 \cdot \cos(18.22 \cdot 4)\\ v &= -0.4422 \text{ m/s} \end{align*}

Finally, using the acceleration equation for simple harmonic motion:

\begin{align*} a &= -A \cdot \omega^2 \cdot\sin(\omega t)\\ &= -3 \cdot 18.22^2 \cdot\sin(18.22 \cdot 4)\\ a &= 5.855 \text{ m/s}^2 \end{align*}

## How to use this simple harmonic motion calculator

This simple harmonic motion calculator requires the following inputs:

• Amplitude $A$ of the oscillation;
• Time $t$ at which we need to calculate; and
• Frequency $f$ or angular frequency $\omega$ of the oscillation.

Using these inputs, the calculator can determine the oscillating particle's motion parameters:

• Displacement $y$;
• Velocity $v$; and
• Acceleration $a$.
Krishna Nelaturu
Amplitude
in
Time
sec
Frequency
Hz
Angular frequency
per s
Displacement
in
Velocity
m/s
Acceleration
m/s²
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