Displacement
Signed position from equilibrium.
Physics · Oscillations & waves
Simple harmonic motion calculator
Find displacement, signed velocity and acceleration at a specified time in ideal harmonic motion. The phase convention is x = A cos(2πft): at t = 0 the body starts at +A with zero velocity.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Velocity
Signed instantaneous velocity.
Acceleration
Restoring acceleration toward equilibrium.
Angular frequency
rad/s
Angular phase rate, in radians per second.
Relation and numerical substitution
x = A cos(2πft)—
Constant amplitude and frequency, no damping, no forcing; phase fixed at zero.
01Formulas and symbols
Formulas used
ω
ω = 2 × π × fAngular phase rate, in radians per second.x
x = A × cos(ω × t)Signed position from equilibrium.v
v = −A × ω × sin(ω × t)Signed instantaneous velocity.a
a = −A × ω² × cos(ω × t)Restoring acceleration toward equilibrium.02Assumptions and limits
Scope of validity
- Constant amplitude and frequency, no damping, no forcing; phase fixed at zero.
03Validation example
Reference numerical case
- A = 0.10 m, f = 1 Hz and t = 0.25 s: ω = 6.283185… rad/s, x = 0, v = −0.6283185… m/s and a = 0.
04References
Technical references
FAQ
Where is the speed greatest?
At equilibrium x = 0; its magnitude reaches Aω. Acceleration is then zero.
How do I find f for a spring or pendulum?
Use the linked natural-frequency or pendulum calculator, then enter that frequency here to examine the motion over time.
05Model and conventions
Find displacement, signed velocity and acceleration at a specified time in ideal harmonic motion. The phase convention is x = A cos(2πft): at t = 0 the body starts at +A with zero velocity.
06Limits of this model
This gives the instantaneous state, not spring stiffness, natural frequency or the response to an external excitation. Large phase arguments can lose floating-point precision.
07Common mistakes
Frequency in Hz is not angular frequency in rad/s. The velocity and acceleration signs describe direction, not their magnitudes.