Oscillations

Simple harmonic motion happens when a restoring force pulls an object back toward equilibrium with a size proportional to its displacement, like a spring's $F = -kx$. The equilibrium position is where the net force is zero, and SHM is a special kind of periodic motion.

The period T is the time for one full cycle, the frequency is $f = \frac{1}{T}$ and the angular frequency is $\omega = 2\pi f$. A mass on a spring has $T = 2\pi\sqrt{\frac{m}{k}}$ and a simple pendulum at small angles has $T = 2\pi\sqrt{\frac{\ell}{g}}$; neither depends on the amplitude.

For SHM, Newton's second law gives $\frac{d^2x}{dt^2} = -\omega^2 x$. Its solution is $x(t) = A\cos(\omega t + \phi)$, and differentiating gives the velocity and acceleration, with maximum speed $A\omega$ and maximum acceleration $A\omega^2$. Pushing a system at its natural frequency causes resonance, which makes the amplitude grow.

In SHM, energy keeps changing form while the total stays the same: kinetic energy is greatest at equilibrium, and potential energy is greatest at the turning points. For a spring–object system the total energy is $E = \frac{1}{2}kA^2$, so a bigger amplitude means more energy but the same period.

A physical pendulum is a rigid object swinging about a fixed pivot. Gravity acting at its center of mass makes a restoring torque, and for small angles ($\sin\theta \approx \theta$) Newton's second law for rotation becomes SHM with $T = 2\pi\sqrt{\frac{I}{mgd}}$, where d is the distance from the pivot to the center of mass. A simple pendulum is the special case of a point mass on a string. In a torsion pendulum, a twisted wire pushes back with a torque proportional to the twist angle.