SHM Core

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

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

$a = -\omega^2 x \quad [M^0 L^1 T^{-2}] \text{ (m/s}^2\text{)}$

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

SHM Energy

$KE = \frac{1}{2}m\omega^2(A^2 - x^2) \quad [M^1 L^2 T^{-2}] \text{ (J)}$

$PE = \frac{1}{2}m\omega^2 x^2 \quad [M^1 L^2 T^{-2}] \text{ (J)}$

$E_{total} = \frac{1}{2}m\omega^2 A^2 = \text{constant} \quad [M^1 L^2 T^{-2}] \text{ (J)}$

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

Oscillating Systems

$T_{spring} = 2\pi\sqrt{\frac{m}{k}} \quad k \text{ in } [M^1 L^0 T^{-2}] \text{ (N/m)}$

$T_{pendulum} = 2\pi\sqrt{\frac{L}{g}} \quad [\text{valid for } \theta < 15°]$

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

Waves

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

$v_{string} = \sqrt{\frac{T}{\mu}} \quad \text{Dim. check: } \sqrt{\frac{[MLT^{-2}]}{[ML^{-1}]}} = [LT^{-1}] \checkmark$

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

Standing Waves & Pipes

$f_n^{open} = \frac{nv}{2L}, \quad n = 1, 2, 3\ldots \quad [M^0 L^0 T^{-1}] \text{ (Hz)}$

$f_n^{closed} = \frac{nv}{4L}, \quad n = 1, 3, 5\ldots \text{ (odd only)}$

Beats & Doppler

$f_{beat} = |f_1 - f_2| \quad [M^0 L^0 T^{-1}] \text{ (Hz)}$

$f' = f\cdot\frac{v \pm v_O}{v \mp v_S} \quad \text{(+ toward for observer, − toward for source)}$