Free · Rotational and periodic motion calculators 🌎

Simple Harmonic Motion — Sine-Phase Model

Calculate displacement, velocity and acceleration for ideal simple harmonic motion with zero initial sine phase.

Calculator inputs

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Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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For amplitude 100 cm and angular frequency 2 rad/s, at time zero the sine-phase model gives displacement 0, velocity 2 m/s and acceleration 0.

y = A sin(ωt); v = Aω cos(ωt); a = −ω²y; ω = 2πf

Enter these known values and leave the other values blank.

This model fixes the sine phase to zero: at time zero displacement is zero and velocity is positive for positive amplitude.
Amplitude is nonnegative. Frequency and angular frequency are positive. The oscillator is ideal and undamped.
Inverse sine and cosine calculations return principal branches, not every time in the oscillation cycle. Displacement magnitude cannot exceed amplitude.

This tool has a fixed zero sine phase. Change the time origin to the positive equilibrium crossing before using it.

For this ideal oscillator, acceleration equals minus angular frequency squared times displacement.

Bookify requires nonnegative amplitude, positive oscillation frequency and displacement within the amplitude.