Waves & Oscillations
Simple Harmonic Motion Calculator
Calculate SHM period, frequency, max velocity, max acceleration and instantaneous position. Formula: x(t) = Acos(ωt).
Period & Frequency
x(t) = A cos(ωt)
Period (T)
—
s
Frequency (f)
—
Hz
v_max
—
m/s
a_max
—
m/s²
Step-by-Step Solution
What is Simple Harmonic Motion?
Simple Harmonic Motion (SHM) occurs when the restoring force on an object is proportional to its displacement from equilibrium and directed toward it. Examples include springs, pendulums (small angle), oscillating molecules and AC circuits. SHM produces a sinusoidal displacement-time graph.
x(t) = A cos(ωt) | v(t) = −Aω sin(ωt) | a(t) = −Aω² cos(ωt)
A = amplitude (m) | ω = angular frequency (rad/s) | T = 2π/ω | f = ω/(2π)
💡 In SHM, energy continuously converts between kinetic and potential forms. At maximum displacement (x = A), all energy is potential. At equilibrium (x = 0), all energy is kinetic. Total energy = ½kA² = constant.
Real-World Applications
Molecular Physics
Atoms in a solid oscillate around equilibrium positions — this is SHM that determines thermal properties.
Musical Instruments
Vibrating strings and air columns in instruments undergo SHM, producing pure musical tones.
LC Circuits
Electrical oscillations in inductors and capacitors are mathematically identical to mechanical SHM.
Seismology
Seismometers measure ground oscillations by detecting SHM of a suspended mass during earthquakes.
Frequently Asked Questions
What is the difference between SHM and periodic motion?
All SHM is periodic (repeating), but not all periodic motion is SHM. SHM requires the restoring force to be proportional to displacement (F = -kx). A bouncing ball is periodic but not SHM because the restoring force is not proportional.
What is angular frequency?
Angular frequency ω = 2πf = 2π/T (rad/s). It represents how fast the angle in the sinusoidal oscillation changes. For a spring: ω = √(k/m). For a pendulum: ω = √(g/L).
How does amplitude affect SHM period?
For ideal SHM, the period is independent of amplitude. A pendulum swings with the same period whether the amplitude is 1° or 10° (for small angles). This isochronous property was discovered by Galileo and made pendulum clocks possible.