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=Asin(ωt)y = A \cdot \sin(\omega t)

Where:

  • yy - Displacement from the equilibrium position;
  • AA - Amplitude of oscillation (maximum displacement);
  • ω\omega - Angular frequency of the oscillation;
  • tt - Time instant at which we want to measure the displacement.

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

ω=2πf\omega = 2 \pi f

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

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

Where vv is the velocity of the oscillating particle.

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

a=ddt(Aωcos(ωt))a=Aω2sin(ωt)a = \frac{d}{dt}\left(A \cdot \omega \cdot \cos(\omega t)\right) \\[1em] a = -A \cdot \omega^2 \cdot\sin(\omega t)

Where aa 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 cm3 \text{ cm} at a frequency of 2.9 Hz2.9 \text{ Hz}, what are its displacement, velocity, and acceleration 4 seconds4 \text{ seconds} into the oscillation?

Given data:

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

To find:
Displacement, velocity, and acceleration

Let's start by finding the angular frequency:

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

Using the displacement formula for simple harmonic motion:

y=Asin(ωt)=3sin(18.224)y=1.76 cm\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:

v=Aωcos(ωt)=318.22cos(18.224)v=0.4422 m/s\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:

a=Aω2sin(ωt)=318.222sin(18.224)a=5.855 m/s2\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 AA of the oscillation;
  • Time tt at which we need to calculate; and
  • Frequency ff or angular frequency ω\omega of the oscillation.

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

  • Displacement yy;
  • Velocity vv; and
  • Acceleration aa.
Krishna Nelaturu
Amplitude
in
Time
sec
Frequency
Hz
Angular frequency
rad
per s
Displacement
in
Velocity
m/s
Acceleration
m/s²
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