SHM Core Equations:

$x = A\sin(\omega t + \phi) \quad [M^0L^1T^0]\text{ m}$

$v = \omega\sqrt{A^2-x^2},\quad v_{max} = A\omega \quad [M^0L^1T^{-1}]\text{ m/s}$

$a = -\omega^2 x,\quad a_{max} = A\omega^2 \quad [M^0L^1T^{-2}]\text{ m/s}^2$

$T = \frac{2\pi}{\omega} \quad [M^0L^0T^1]\text{ s}$

SHM Energy:

$E = KE + PE = \frac{1}{2}m\omega^2 A^2 \quad [M^1L^2T^{-2}]\text{ J}$

$\text{KE} = \text{PE} \text{ at } x = \frac{A}{\sqrt{2}}$

Systems:

$T_{spring} = 2\pi\sqrt{\frac{m}{k}},\quad T_{pendulum} = 2\pi\sqrt{\frac{L}{g}}$

$k_{series}: \frac{1}{k_{eff}}=\frac{1}{k_1}+\frac{1}{k_2} \qquad k_{parallel}: k_{eff}=k_1+k_2$

Waves:

$v = f\lambda = \frac{\omega}{k} \quad [M^0L^1T^{-1}]\text{ m/s}$

$v_{string} = \sqrt{\frac{T}{\mu}},\quad \left[\sqrt{\frac{MLT^{-2}}{ML^{-1}}}=LT^{-1}\right]\checkmark$

$v_{sound} = \sqrt{\frac{\gamma RT}{M}} \propto \sqrt{T_K}$

Pipes & Beats:

$f_n^{open} = \frac{nv}{2L},\;n=1,2,3\ldots \qquad f_n^{closed} = \frac{nv}{4L},\;n=1,3,5\ldots$

$f_{beat} = |f_1 - f_2|$

Doppler:

$f' = f\cdot\frac{v \pm v_O}{v \mp v_S}$