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.
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.