Simple Harmonic Motion Calculator

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.

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.

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

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:

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

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

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

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

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:

Using the displacement formula for simple harmonic motion:

Using the velocity equation for simple harmonic motion:

Finally, using the acceleration equation for simple harmonic motion:

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 $a$.