Oscillations

Simple harmonic motion (SHM) happens when a restoring force pulls an object back toward its equilibrium position with a strength proportional to how far it's displaced, like an ideal spring with F = −kx. A pendulum swinging through small angles is very close to SHM because the restoring torque on it is nearly proportional to its angle.

The period T is the time for one full cycle and the frequency f is the number of cycles per second, so T = 1/f. A mass on a spring has T = 2π√(m/k) and a pendulum at small angles has T = 2π√(L/g), so a pendulum's period doesn't depend on its mass, and neither period depends on the amplitude.

An object in SHM has position x = A cos(2πft) (or the sine version), so its position, velocity and acceleration graphs are all sinusoidal. At equilibrium its speed is greatest and its acceleration is zero; at the turning points its speed is zero and its acceleration is greatest, pointing back toward equilibrium.

An oscillator's total mechanical energy stays constant as it shifts between kinetic and potential: at the turning points the kinetic energy is zero, and at equilibrium the kinetic energy is greatest and the potential energy is lowest. For a mass on a spring the total energy is ½kA², so doubling the amplitude makes the total energy four times as large.