Simple Harmonic Motion

Position, velocity, and acceleration of an oscillator

x(t) = A·cos(ωt + φ)
Position x(t)
0.5403 m
Velocity v(t)
-1.6829 m/s
Acceleration a(t)
-2.1612 m/s²
Period (T)
3.1416 s
Frequency (f)
0.3183 Hz
Total phase (ωt + φ)
57.3°
Max speed (vₘₐₓ)
2 m/s
Max acceleration (aₘₐₓ)
4 m/s²

Position follows x(t) = A·cos(ωt + φ); velocity and acceleration are its first and second derivatives, with a(t) = −ω²·x(t).

Simple Harmonic MotionDisplacement curve x(t) = A·cos(ωt + φ) with amplitude 1 m and period 3.1416 s; the marker shows position 0.5403 m at time 0.5 s.+A−Aequilibriumnowx = 0.5403 m

About this calculator

The Simple Harmonic Motion Calculator finds the instantaneous position, velocity, and acceleration of an oscillator described by x(t) = A·cos(ωt + φ). Enter the amplitude, angular frequency, phase angle, and time to get the displacement x(t) = A·cos(ωt + φ), velocity v(t) = −Aω·sin(ωt + φ), and acceleration a(t) = −ω²·x(t), along with the period, frequency, maximum speed, and maximum acceleration. Every value is computed with 30-digit precision and the curve shows where the oscillator is at the chosen moment. Useful for physics homework, springs, pendulums in the small-angle limit, and any oscillating system.

Common examples

  • A = 1 m, ω = 1 rad/s, φ = 0°, t = 0 s → x = 1 m, v = 0 m/s, a = −1 m/s²
  • A = 2 m, ω = 3 rad/s, φ = 0°, t = 0 s → x = 2 m, v = 0 m/s, a = −18 m/s²
  • A = 1 m, ω = 2 rad/s, φ = 90°, t = 0 s → x = 0 m, v = −2 m/s, a = 0 m/s²
  • ω = 2π rad/s → period T = 1 s, frequency f = 1 Hz
  • A = 0.5 m, ω = 4 rad/s → max speed 2 m/s, max acceleration 8 m/s²