Key Equations

Key Equations

Relationship between frequency and period	$f=\frac{1}{T}$
$\text{Position in SHM with}\varphi =0.00$	$x\left(t\right)=A\text{cos}\left(\omega t\right)$
General position in SHM	$x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$
General velocity in SHM	$v\left(t\right)=\text{−}A\omega \text{sin}\left(\omega t+\varphi \right)$
General acceleration in SHM	$a\left(t\right)=\text{−}A{\omega }^2\text{cos}\left(\omega t+\varphi \right)$
Maximum displacement (amplitude) of SHM	$x_{\text{max}}=A$
Maximum velocity of SHM	$\left|v_{\text{max}}\right|=A\omega$
Maximum acceleration of SHM	$\left|a_{\text{max}}\right|=A{\omega }^2$
Angular frequency of a mass-spring system in SHM	$\omega =\sqrt{\frac{k}{m}}$
Period of a mass-spring system in SHM	$T=2\pi \sqrt{\frac{m}{k}}$
Frequency of a mass-spring system in SHM	$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
Energy in a mass-spring system in SHM	$E_{\text{Total}}=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2=\frac{1}{2}k{A}^2$
The velocity of the mass in a spring-mass system in SHM	$v=\pm \sqrt{\frac{k}{m}\left(A^2-x^2\right)}$
The x-component of the radius of a rotating disk	$x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$
The x-component of the velocity of the edge of a rotating disk	$v\left(t\right)=\text{−}{v}_{\text{max}}\text{sin}\left(\omega t+\varphi \right)$
The x-component of the acceleration of the edge of a rotating disk	$a\left(t\right)=\text{−}{a}_{\text{max}}\text{cos}\left(\omega t+\varphi \right)$
Force equation for a simple pendulum	$\frac{d^2\theta }{d{t}^2}=-\frac{g}{L}\theta$
Angular frequency for a simple pendulum	$\omega =\sqrt{\frac{g}{L}}$
Period of a simple pendulum	$T=2\pi \sqrt{\frac{L}{g}}$
Angular frequency of a physical pendulum	$\omega =\sqrt{\frac{mgL}{I}}$
Period of a physical pendulum	$T=2\pi \sqrt{\frac{I}{mgL}}$
Period of a torsional pendulum	$T=2\pi \sqrt{\frac{I}{\kappa }}$
Newton’s second law for harmonic motion	$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=0$
Solution for underdamped harmonic motion	$x\left(t\right)=A_0{e}^{-\frac{b}{2m}t}\text{cos}\left(\omega t+\varphi \right)$
Natural angular frequency of a mass-spring system	${\omega }_0=\sqrt{\frac{k}{m}}$
Angular frequency of underdamped harmonic motion	$\omega =\sqrt{{\omega }_0^2-{\left(\frac{b}{2m}\right)}^2}$
Newton’s second law for forced, damped oscillation	$\text{−}kx-b\frac{dx}{dt}+F_o\text{sin}\left(\omega t\right)=m\frac{d^{2}x}{d{t}^2}$
Solution to Newton’s second law for forced, damped oscillations	$x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$
Amplitude of system undergoing forced, damped oscillations	$A=\frac{F_o}{\sqrt{m^2{\left({\omega }^2-{\omega }_o^2\right)}^2+b^2{\omega }^2}}$