Oscillations

Periodic: repeats identically after a fixed period $T$ (frequency $\nu = 1/T$, in Hz). Oscillatory: periodic and to-and-fro about a fixed mean position. Every oscillation is periodic; not every periodic motion (e.g. uniform circular motion) is oscillatory.

$\omega = 2\pi/T = 2\pi\nu$. Speed max ($A\omega$) at the mean position; acceleration max ($A\omega^2$) at the extremes.

SHM = projection of uniform circular motion onto a diameter.

Any linear restoring force $F=-kx$ guarantees SHM with $\omega=\sqrt{k/m}$, whatever the system.

Constant for all time. $K$ and $U$ each oscillate at twice the frequency of $x(t)$, out of step: $U$ peaks at $x=\pm A$, $K$ peaks at $x=0$. They cross at $x=\pm A/\sqrt2$.

Pendulum derivation needs $\sin\theta\approx\theta$, valid only for small $\theta$ (roughly under 10°).

1. Spring-block: period is independent of amplitude — always, exactly, since $F=-kx$ is exactly linear at any amplitude within the linear range.
2. Simple pendulum, large angle: isochronism breaks down — period creeps up with amplitude, because $\sin\theta$ falls measurably below $\theta$ and the restoring force is only approximately linear, and only at small amplitude.

Damping force $F_d=-bv$; lowered frequency $\omega'=\sqrt{k/m-b^2/4m^2}$. Energy decays as $E(t)=\tfrac12kA^2e^{-bt/m}$ — faster for larger $b$.

Driven at frequency $\omega_d$, a damped oscillator settles into steady oscillation at $\omega_d$ itself, not its natural frequency $\omega_0=\sqrt{k/m}$. As $\omega_d \to \omega_0$, amplitude grows sharply — resonance, limited only by damping.