SHM Displacement Calculator
Enter amplitude, angular frequency, time and phase to compute SHM displacement x=A·cos(ωt+φ), plus velocity v=−Aω·sin(ωt+φ), acceleration a=−ω²x and period T=2π/ω. A=0.1, ω=2, t=1, φ=0 → x≈−0.0414 m.
Input Data
Results
At a glance:Simple harmonic motion (SHM): when the restoring force is proportional to the displacement (Hooke's law, F=−kx), the motion follows x(t)=A·cos(ωt+φ), where x is displacement, A amplitude (m), ω angular frequency (rad/s), φ phase (rad), t time (s). Derived by differentiating: velocity v=dx/dt=−Aω·sin(ωt+φ), acceleration a=dv/dt=−ω²·x (a is always opposite to and proportional to x — the hallmark of SHM). Period T=2π/ω, frequency f=ω/(2π). A and ω are independent: A sets the swing size, ω the speed (decided by stiffness/mass). History: Galileo noticed the pendulum's isochronism in 1583; Huygens made the pendulum clock in 1656; Newton formally derived SHM in the 1680s. Example: A=0.1 m, ω=2 rad/s, t=1 s, φ=0 → x=0.1×cos(2)=0.1×(−0.416)=−0.0416 m, v=−0.1×2×sin(2)=−0.2×0.909=−0.182 m/s, a=−2²×(−0.0416)=0.166 m/s². Applications: (1) mass-spring and pendulum (small angle); (2) LC circuits (charge oscillates); (3) waves and acoustics; (4) vibration and seismology; (5) clock escapements.
Formula
Displacement: x = A·cos(ωt + φ)
Velocity: v = −Aω·sin(ωt + φ)
Acceleration: a = −ω²·x
Period & frequency: T = 2π/ω, f = ω/(2π)
Energy: E = ½kA² = ½mω²A² (constant)
$$x(t) = A\cos(\omega t + \phi), \quad v = -A\omega\sin(\omega t+\phi), \quad a = -\omega^2 x$$How to Use
- Enter amplitude A (m) and angular frequency ω (rad/s).
- Enter time t (s) and phase φ (rad).
- The tool gives x=A·cos(ωt+φ), velocity, acceleration and period T=2π/ω.
Case Studies
Mass-spring and pendulum
Mass-spring: ω=√(k/m). k=100, m=1 → ω=10 rad/s, T=0.628 s.
Pendulum: ω=√(g/L), g=9.81, L=1 → ω=3.13 rad/s, T=2.01 s (Hong Kong's clock towers).
Both are SHM for small amplitude; large pendulum angle needs the large-angle correction.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.