Summary

Periodic motion: motion repeating after a fixed time $T$ (the period); frequency $f = 1/T$ (Hz); angular frequency $\omega = 2\pi f = 2\pi/T$ (rad/s). A function $f(t)$ is periodic if $f(t+T)=f(t)$ for all $t$; sine/cosine are the simplest periodic functions, and any periodic function can be built from sums of them. S.H.M.: defined by a restoring force $F=-kx$, giving $d^2x/dt^2 + \omega^2x = 0$ with $\omega^2=k/m$, and solution $x(t)=A\cos(\omega t+\phi)$; $A$ = amplitude, $(\omega t+\phi)$ = phase, $\phi$ = phase constant; the period $T=2\pi\sqrt{m/k}$ is independent of amplitude. Velocity and acceleration: $v=-A\omega\sin(\omega t+\phi)$, $v_{\max}=A\omega$ at $x=0$; $a=-\omega^2x$, $a_{\max}=A\omega^2$ at $x=\pm A$. Springs: force constant $k$ (N/m) from $F=-kx$; series combination $1/k_s=1/k_1+1/k_2$ (softer); parallel combination $k_p=k_1+k_2$ (stiffer). Energy in S.H.M.: $K(x)=\tfrac12m\omega^2(A^2-x^2)$, $U(x)=\tfrac12m\omega^2x^2=\tfrac12kx^2$, total $E=K+U=\tfrac12kA^2$, constant at every $x$; $E$ is fully kinetic at $x=0$ and fully potential at $x=\pm A$. Simple pendulum: for small $\theta$, $d^2\theta/dt^2=-(g/L)\theta$, giving $T=2\pi\sqrt{L/g}$, independent of mass and amplitude. Loaded spring: gravity only shifts the equilibrium point; the restoring force about the new equilibrium is still $-kx$, giving the same $T=2\pi\sqrt{m/k}$ as a horizontal spring, independent of $g$. Free oscillations: at the system's own natural frequency $\omega_0$, amplitude ideally constant. Damped oscillations: amplitude decreases with time as energy is lost to resistive forces. **Forced osci …