Physics · Interactive
Simple Harmonic Motion
Simple harmonic motion traces a smooth sine wave over time. Drag the amplitude and period below to reshape the oscillation and watch the frequency and angular frequency update live. Free to use, and exportable into your slides.
Drag the amplitude and period to reshape the wave. Open fullscreen ↗
Use it in your lesson
Drop this into your slides or hand it to students. Free downloads carry a small figviz.io watermark; spend 1 credit to export a clean, high-resolution version.
Remove watermark · high-res · 1 credit each
What is simple harmonic motion?
Simple harmonic motion, or SHM, is a repeating back-and-forth motion in which the restoring force is proportional to how far the object is from its resting point and always pushes it back toward that point. A mass on a spring and a small-swing pendulum are classic examples. Plotting the displacement against time gives a smooth sine wave. Drag the amplitude and period in the grapher above to reshape that wave.
Amplitude, period, and frequency
The amplitude A is the greatest distance from the resting point, so it sets the height of the wave. The period T is the time for one complete cycle, measured in seconds, and it sets how stretched out the wave is. The frequency f is the number of cycles per second, measured in hertz, and it is simply one divided by the period. The panel reports all three as you drag the sliders.
The displacement equation and angular frequency
Displacement in SHM is written y(t) = A·sin(ωt), where ω is the angular frequency in radians per second. Angular frequency connects to the period by ω = 2π / T, and to the frequency by ω = 2πf. Because a full cycle is 2π radians, ω tells you how fast the phase advances. The grapher shows the equation with the current values so you can match each symbol to the curve.
Why the graph is a sine wave
The defining feature of SHM is that acceleration is proportional to the negative of displacement, which is exactly the property of the sine and cosine functions. Solving that relationship gives sinusoidal motion, so the displacement-time graph is always a smooth, repeating sine wave. Changing the amplitude scales the wave vertically, while changing the period stretches or compresses it horizontally without changing its shape.
Frequently asked questions
What is simple harmonic motion?
Simple harmonic motion is oscillation in which the restoring force is proportional to displacement and directed back toward equilibrium. Its displacement-time graph is a sine wave. A mass on a spring is a standard example.
What are amplitude, period, and frequency?
Amplitude is the maximum distance from equilibrium, period is the time for one full cycle in seconds, and frequency is the number of cycles per second in hertz. Frequency is one divided by the period.
What is the equation for simple harmonic motion?
A common form is y(t) = A·sin(ωt), where A is the amplitude and ω is the angular frequency in radians per second. It gives the displacement y at any time t.
What is angular frequency?
Angular frequency ω is how fast the phase of the oscillation advances, in radians per second. It relates to the period by ω = 2π / T and to the frequency by ω = 2πf.
Why is the SHM graph a sine wave?
In SHM the acceleration is proportional to the negative of the displacement, which is the defining property of sine and cosine. Solving that relationship produces sinusoidal motion, so the graph is always a sine wave.
How do I use this simple harmonic motion grapher?
Drag the amplitude and period sliders. The sine wave redraws instantly and the panel shows the amplitude, period, frequency, and angular frequency, along with the displacement equation. You can also export the graph.