Simple Harmonic Motion Calculator
Calculate the period, frequency, max velocity, max acceleration, and total energy of a spring-mass system undergoing simple harmonic motion.
Period
Simple harmonic motion (SHM) occurs when a restoring force is proportional to displacement from equilibrium: F = −kx.
Key formulas:
- ω = √(k/m) — angular frequency (rad/s)
- T = 2π/ω = 2π√(m/k) — period (seconds)
- f = 1/T — frequency (Hz)
- v_max = Aω — maximum velocity (at equilibrium)
- a_max = Aω² — maximum acceleration (at extremes)
- E_total = ½kA² — total mechanical energy (constant)
Where m = mass, k = spring constant, A = amplitude.
Energy conversion in SHM:
- At maximum displacement (x = ±A): all energy is potential (E_p = ½kA²), velocity = 0
- At equilibrium (x = 0): all energy is kinetic (E_k = ½mv²_max), acceleration = 0
- At any point: E_p + E_k = ½kA² = constant
Real pendulum: For a simple pendulum of length L: ω = √(g/L), T = 2π√(L/g) A 1-meter pendulum on Earth has T ≈ 2.006 seconds — this was used historically to define the unit of length.
SHM in everyday life:
- Sound waves (pressure oscillations)
- AC electrical current (voltage oscillates sinusoidally)
- Vibrating guitar strings and tuning forks
- Atomic vibrations in molecules (infrared spectroscopy)
- Seismic waves (simplified)