Simple Harmonic Motion Formulas
The complete set of simple harmonic motion equations: position, velocity, acceleration, period, frequency, and energy for springs and pendulums.
The Formulas
v(t) = −Aω sin(ωt + φ)
a(t) = −Aω² cos(ωt + φ)
ω = √(k/m) = 2πf
T = 2π/ω = 2π√(m/k)
E_total = ½kA²
Simple harmonic motion (SHM) occurs when a restoring force is proportional to displacement from equilibrium. The classic example is a spring: F = −kx. The negative sign means the force always points back toward the center. This same pattern appears in pendulums, electrical LC circuits, sound waves, molecular vibrations, and countless other systems.
Calculator
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| x(t) | Position at time t | meters (m) |
| A | Amplitude — maximum displacement from equilibrium | meters (m) |
| ω | Angular frequency | rad/s |
| φ | Phase constant — depends on initial conditions | radians |
| k | Spring constant (stiffness) | N/m |
| m | Mass of the oscillating object | kg |
| T | Period — time for one complete oscillation | seconds (s) |
| f | Frequency = 1/T | Hz |
| E_total | Total mechanical energy (constant throughout motion) | Joules (J) |
Example 1 — Spring-Mass System
A 2 kg mass is attached to a spring with k = 50 N/m. The amplitude is 0.1 m. Find the period, frequency, max velocity, and max acceleration.
ω = √(k/m) = √(50/2) = √25 = 5 rad/s
T = 2π/ω = 2π/5 = 1.257 seconds
f = 1/T = 0.796 Hz
v_max = Aω = 0.1 × 5 = 0.5 m/s (occurs at equilibrium position)
a_max = Aω² = 0.1 × 25 = 2.5 m/s² (occurs at maximum displacement)
E_total = ½kA² = ½ × 50 × 0.01 = 0.25 J
Example 2 — Simple Pendulum
For a simple pendulum of length L, the period is T = 2π√(L/g). Find the period of a 1-meter pendulum on Earth (g = 9.81 m/s²).
T = 2π × √(1/9.81)
T = 2π × √0.10194
T = 2π × 0.31928
T ≈ 2.006 seconds — a 1-meter pendulum swings back and forth in almost exactly 2 seconds. This is why it was historically used to define units of length!
When to Use It
Use the SHM formulas when:
- Analyzing spring-mass systems in mechanics problems
- Designing vibration isolators, shock absorbers, and seismic instruments
- Modeling pendulum clocks, metronomes, and oscillators
- Studying molecular vibrations and infrared spectroscopy
- Analyzing LC oscillator circuits in electronics (ω = 1/√(LC))
The total energy E = ½kA² is constant: at maximum displacement, all energy is potential (½kx²). At equilibrium (x = 0), all energy is kinetic (½mv²). Energy continuously converts between kinetic and potential forms — this is the hallmark of ideal SHM.