Simple Harmonic Motion Calculator
Calculate period, frequency, displacement, velocity and acceleration for an ideal oscillator. Enter frequency directly or use a mass–spring or small-angle pendulum model.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Find angular frequency from the chosen model, then use the cosine displacement convention. Differentiate for signed velocity and acceleration. Initial phase determines the state at t=0.
x=A cos(ωt+φ); v=−Aω sin(ωt+φ); a=−ω²x; T=2π/ω
Worked example
For A=0.05 m, f=1 Hz and phase zero, t=0.25 s gives x≈0, v≈−0.314159 m/s and a≈0. The period is 1 s.
Assumptions and input details
Undamped linear oscillation with constant parameters and no driving force. Mass–spring uses ω=√(k/m), with positive mass and spring constant. A simple pendulum uses ω≈√(g/L), with positive length/gravity, and displacement means arc length from equilibrium. Its arc amplitude/length must be ≤0.1 rad. Initial phase is within ±360° or ±2π rad. Time is nonnegative up to 1e9 s and within 10 million cycles for phase precision; frequency is 1e−12–1e12 Hz. The chart shows one period starting at time zero, independent of the selected evaluation time.
References: OpenStax simple harmonic motion · OpenStax pendulums
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
The 19 mass definitions distinguish avoirdupois and troy measures, metric and regional measures. Japanese kin is 600 g; mainland Chinese shijin is 500 g.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
SI prefixes are explicit: for example, milli means one thousandth, micro one millionth and kilo one thousand. Changing a unit changes the interpretation of the entered number.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Find applied spring force, extension or spring constant in the elastic range. Distinguish applied and restoring forces and calculate stored elastic energy.
Convert rpm, angular speed and tangential rim speed using a radius or diameter. Find frequency, period and centripetal acceleration with explicit units.
Calculate wave speed, frequency or wavelength from the other two. Convert units, find the period and view an illustrative normalized wave.
Reserved advertising space.