Simple Harmonic Motion Calculator
Compute the displacement, velocity, or acceleration of an oscillating object in simple harmonic motion at any point in time. Used in physics and engineering to analyse springs, pendulums, and wave systems.
About this calculator
Simple harmonic motion (SHM) describes oscillation where the restoring force is proportional to displacement. Three quantities define the motion at any instant. Displacement: x(t) = A·cos(ωt + φ), where A is amplitude (m), ω is angular frequency (rad/s), t is time (s), and φ is the phase constant (rad). Velocity is the time derivative: v(t) = −A·ω·sin(ωt + φ). Acceleration is the second derivative: a(t) = −A·ω²·cos(ωt + φ). Note that acceleration equals −ω²·x(t), confirming the defining SHM relationship. Angular frequency relates to period T by ω = 2π/T and to natural frequency f by ω = 2πf. The phase constant φ sets the initial conditions of the oscillation.
How to use
A mass on a spring has amplitude A = 0.05 m, angular frequency ω = 4 rad/s, and phase constant φ = 0 rad. Find displacement and velocity at t = 0.5 s. Displacement: x = 0.05·cos(4·0.5 + 0) = 0.05·cos(2) = 0.05·(−0.4161) ≈ −0.0208 m. Velocity: v = −0.05·4·sin(4·0.5 + 0) = −0.2·sin(2) = −0.2·0.9093 ≈ −0.1819 m/s. Acceleration: a = −0.05·16·cos(2) = −0.8·(−0.4161) ≈ 0.3329 m/s². Enter these values, select the calculation type, and read the result directly.
Frequently asked questions
What is angular frequency in simple harmonic motion and how does it relate to period?
Angular frequency ω (omega) measures how rapidly an oscillator completes its cycle, expressed in radians per second. It relates to the period T (seconds per cycle) by ω = 2π/T, and to ordinary frequency f (cycles per second) by ω = 2πf. For a mass-spring system, ω = √(k/m), where k is the spring constant and m is mass. A higher angular frequency means faster oscillations and a shorter period. Knowing ω lets you immediately calculate the full time evolution of displacement, velocity, and acceleration.
How does the phase constant affect simple harmonic motion?
The phase constant φ (phi) sets the initial state of the oscillator at t = 0. When φ = 0, the object starts at maximum positive displacement (x = A). When φ = π/2, the object starts at x = 0 and moves in the negative direction. Physically, φ encodes the starting conditions: initial displacement and initial velocity together uniquely determine φ. In practice, φ is often set to zero when timing begins at a turning point, and to π/2 when timing begins as the object passes through equilibrium.
What is the relationship between displacement and acceleration in simple harmonic motion?
In SHM, acceleration is always directly proportional to displacement but opposite in sign: a(t) = −ω²·x(t). This is the defining equation of simple harmonic motion and means the restoring force always points back toward the equilibrium position. The proportionality constant is ω², so a stiffer spring (higher ω) produces greater acceleration for the same displacement. This relationship also means that wherever displacement is maximum, acceleration is maximum, and wherever the object passes through equilibrium (x = 0), acceleration is zero but velocity is at its peak.