Simple harmonic motion
SHM equationsoscillation formulasspring periodpendulum periodmass on a spring
Displacement, maximum velocity and maximum acceleration in an oscillation, with the two periods and the restoring force that generate them.
Hooke's Law
Restoring force of an ideal spring, proportional to its displacement from rest.
Period of a Spring-Mass Oscillator
Period of a mass bouncing on a spring, set only by the mass and the spring stiffness.
Simple Pendulum Period
Period of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).
SHM Displacement at Time t
Displacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular frequency ω released from full stretch.
SHM Maximum Velocity
Maximum speed of a simple harmonic oscillator, reached at the equilibrium point, equal to amplitude times angular frequency.
SHM Maximum Acceleration
Maximum acceleration of a simple harmonic oscillator, reached at the turning points where the restoring force is largest.
Elastic Potential Energy
Energy stored in an ideal spring displaced x from its rest length.
How they fit together
Simple harmonic motion is what you get whenever the restoring force is proportional to the displacement and points back toward the middle — Hooke's law, in other words. That one condition forces the motion to be a sine wave, which is why displacement, velocity and acceleration all share the same ω and differ only by a quarter cycle: velocity peaks at the centre where displacement is zero, acceleration peaks at the extremes where the spring is most stretched.
Start from the period. The spring formula depends on mass and stiffness; the pendulum formula depends on length and g and — this is the part people refuse to believe — not on the mass of the bob at all. Once you have T you have ω = 2π/T, and every other quantity follows from amplitude and ω. Two traps: the pendulum result is only good for small swings, roughly under 15°, because it quietly replaces sin θ with θ; and calculators must be in radians for the displacement-at-time formula, since ωt is an angle in radians whatever the problem's other units say.