Key Equations

Relationship between frequency and period	$f = \frac{1}{T}$
$Position in SHM with ϕ = 0.00$	$x (t) = A cos (ω t)$
General position in SHM	$x (t) = A cos (ω t + ϕ)$
General velocity in SHM	$v (t) = − A ω sin (ω t + ϕ)$
General acceleration in SHM	$a (t) = − A ω^{2} cos (ω t + ϕ)$
Maximum displacement (amplitude) of SHM	$x_{max} = A$
Maximum velocity of SHM	$\| v_{max} \| = A ω$
Maximum acceleration of SHM	$\| a_{max} \| = A ω^{2}$
Angular frequency of a mass-spring system in SHM	$ω = \sqrt{\frac{k}{m}}$
Period of a mass-spring system in SHM	$T = 2 π \sqrt{\frac{m}{k}}$
Frequency of a mass-spring system in SHM	$f = \frac{1}{2 π} \sqrt{\frac{k}{m}}$
Energy in a mass-spring system in SHM	$E_{Total} = \frac{1}{2} k x^{2} + \frac{1}{2} m v^{2} = \frac{1}{2} k A^{2}$
The velocity of the mass in a spring-mass system in SHM	$v = \pm \sqrt{\frac{k}{m} (A^{2} - x^{2})}$
The x-component of the radius of a rotating disk	$x (t) = A cos (ω t + ϕ)$
The x-component of the velocity of the edge of a rotating disk	$v (t) = − v_{max} sin (ω t + ϕ)$
The x-component of the acceleration of the edge of a rotating disk	$a (t) = − a_{max} cos (ω t + ϕ)$
Force equation for a simple pendulum	$\frac{d^{2} θ}{d t^{2}} = - \frac{g}{L} θ$
Angular frequency for a simple pendulum	$ω = \sqrt{\frac{g}{L}}$
Period of a simple pendulum	$T = 2 π \sqrt{\frac{L}{g}}$
Angular frequency of a physical pendulum	$ω = \sqrt{\frac{m g L}{I}}$
Period of a physical pendulum	$T = 2 π \sqrt{\frac{I}{m g L}}$
Period of a torsional pendulum	$T = 2 π \sqrt{\frac{I}{κ}}$
Newton’s second law for harmonic motion	$m \frac{d^{2} x}{d t^{2}} + b \frac{d x}{d t} + k x = 0$
Solution for underdamped harmonic motion	$x (t) = A_{0} e^{- \frac{b}{2 m} t} cos (ω t + ϕ)$
Natural angular frequency of a mass-spring system	$ω_{0} = \sqrt{\frac{k}{m}}$
Angular frequency of underdamped harmonic motion	$ω = \sqrt{ω_{0}^{2} - {(\frac{b}{2 m})}^{2}}$
Newton’s second law for forced, damped oscillation	$− k x - b \frac{d x}{d t} + F_{o} sin (ω t) = m \frac{d^{2} x}{d t^{2}}$
Solution to Newton’s second law for forced, damped oscillations	$x (t) = A cos (ω t + ϕ)$
Amplitude of system undergoing forced, damped oscillations	$A = \frac{F_{o}}{\sqrt{m^{2} {(ω^{2} - ω_{o}^{2})}^{2} + b^{2} ω^{2}}}$