Linear SHM

$\omega_0 = \sqrt{\frac{k}{m}}$

$\omega_0 = \sqrt{\frac{g}{l}}$

$\omega_0 = 2 \pi f_0 = \frac{2 \pi}{T}$

$T = \frac{1}{f_0}$

$x(t) = x_0 \cos(\omega_0 t + \phi)$

$\phi = 2\pi \frac{\Delta t}{T}$

$E_T = \frac{1}{2} m v(t)^2 + \frac{1}{2} k x(t)^2 =\, \frac{1}{2} k x_0^2$

$E_T = \frac{1}{2} m v(t)^2 + mgh$

$F_{restoring}\ \propto\ -x(t) $ (restoring force is proportional to negative displacement)
$E_p = \frac{1}{2} \beta x(t)^2$ (Potential energy is quadratic)