The simple harmonic motion calculator computes oscillation properties for spring-mass systems and simple pendulums. SHM occurs whenever a restoring force is proportional to displacement. Key results include period, frequency, angular frequency, and maximum energy.
SHM Calculator
System Type
Step-by-Step
How to Use the Simple Harmonic Motion Calculator
Simple harmonic motion is periodic oscillation where the restoring force is proportional to displacement. This calculator handles two classic SHM systems: the spring-mass system and the simple pendulum.
Step 1: Choose System Type
Select "Spring-Mass" for a mass attached to a spring, or "Simple Pendulum" for a bob on a string under gravity. The formulae differ but the underlying physics is the same.
Spring-Mass Example
A 0.5 kg mass on a spring with k = 50 N/m: Period T = 2π√(0.5/50) = 2π × 0.1 = 0.628 s. Frequency f = 1/T = 1.59 Hz. If amplitude A = 0.1 m, max energy = ½ × 50 × 0.01 = 0.25 J.
Pendulum Example
A 1.0 m pendulum on Earth (g = 9.81 m/s²): T = 2π√(1/9.81) = 2.006 s. A grandfather clock pendulum is typically about 0.993 m for exactly 1-second half-swing (2-second period). The pendulum period doesn't depend on mass or amplitude (for small angles).
Unit Toggle
Switch between metric (kg, m, N/m) and imperial (lb, ft, lb/ft) units. Gravity defaults to 9.81 m/s² (Earth), but you can change it to 1.62 m/s² for the Moon or 3.72 m/s² for Mars.
FAQ
What is simple harmonic motion?
Simple harmonic motion (SHM) is periodic motion where the restoring force is proportional to displacement from equilibrium. Examples include a mass on a spring, a pendulum (small angles), and an LC circuit. The motion is sinusoidal: x(t) = A·cos(ωt + φ).
What is the formula for a spring-mass period?
The period of a spring-mass system is T = 2π√(m/k), where m is mass in kg and k is the spring constant in N/m. Frequency is f = 1/T. Note the period does not depend on amplitude — this is a key property of SHM.
What is the formula for a simple pendulum?
For a simple pendulum (small angle approximation, θ < 15°): T = 2π√(L/g), where L is the string length in meters and g = 9.81 m/s². The period depends only on length and gravity, not on mass or amplitude.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
What is the energy stored in a spring?
The potential energy stored in a compressed/stretched spring is E = ½kA², where k is the spring constant and A is the amplitude. At maximum displacement, all energy is potential; at equilibrium, all energy is kinetic.