Simple Harmonic Motion

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

Period = $T$ Frequency = $f$

Angular frequency $w=\frac{2\pi }{T}$ or $w=2\pi f$

Period: $T=\frac{1}{f}=2\pi \sqrt{\frac{m}{k}}$

Maximum velocity of a wave $=v_{max}=\frac{2\pi A}{T}=2\pi fA=wA$

Maximum acceleration of a wave $=-w^{2}A=(2\pi f{)}^{2}A$

General formulas:

$x(t)=A\cos (wt+{\varphi }_0)$

$v_x(t)=\frac{dx}{dt}=-wA\sin (wt+{\varphi }_0)=-v_{max}\sin (wt+{\varphi }_0)$

$a_x(t)=\frac{d^{2}x}{d{t}^2}=-w^{2}A\cos (wt)$

Energy: $E=K+U=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2$

Traveling Waves

Linear Density

$\mu =\frac{m}{L}$

$T_s=\text{tension}$

Wave speed: $v=f\lambda =\sqrt{\frac{T_s}{\mu }}=\frac{w}{k}$

Note: $v=f\lambda \text{ is from }v=\frac{\text{distance}}{\text{time}}$

Sound

Velocity

$v_{sound}=\sqrt{\frac{B}{\rho }}$

$B=\text{Bulk moduli}Pa$

$\rho =\text{density}kg/m^3$

Dry air at 20degC $\approx 343m/s$

Beat Frequency = $|f_1-f_2|$

Phase

${\varphi }_1=k{x}_1-wt+{\varphi }_0$

${\varphi }_2=k{x}_1-wt+{\varphi }_0$

Phase difference $\Delta \varphi =2\pi \frac{\Delta X}{\lambda }$

Superposition

$D(x,t)=a\sin (kt-\omega t)+a\sin (kt+\omega t)=sin(\alpha )\cos (\beta )\pm \cos (\alpha )\sin (\beta )$

or

$D(x,y)=2a\sin (kx)cos(\omega t)$

$f_m=\frac{v}{{\lambda }_m}=\frac{v}{2L/m}=m\frac{v}{2L}m=1,2,3,\mathrm{4...}$

Fundamental frequency: $f_1=\frac{v}{2L}$

Allowed frequencies: $f_m=m{f}_{1}m=1,2,3,4$

$m=1$	${\lambda }_1=\frac{2L}{1}$	$f_1=\frac{v}{2L}$
$m=2$	${\lambda }_2=\frac{2L}{2}$	$f_2=2\frac{v}{2L}$
$m=3$	${\lambda }_3=\frac{2L}{3}$	$f_3=3\frac{v}{2L}$

Open-open or closed-closed tubes:

$m=1,2,3,\mathrm{4...}$

${\lambda }_m=\frac{2L}{m}$

$f_m=m\frac{v}{2L}=m{f}_1$

Open-closed tubes:

$m=1,3,5,7$

${\lambda }_m=\frac{4L}{m}$

$f_m=m\frac{v}{4L}=m{f}_1$

Maximum interference

Maximum constructive:

$\Delta \varphi =2\pi \frac{\Delta x}{\lambda }+\Delta {\varphi }_0=m\cdot 2\pi \text{rad},m=0,1,2,\mathrm{3...}$

or

$\Delta x=|x_2-x_1|=n\lambda n=0,1,2,...$

Maximum destructive:

$\Delta \varphi =2\pi \frac{\Delta x}{\lambda }+\Delta {\varphi }_0=(m+\frac{1}{2})\cdot 2\pi \text{rad},m=0,1,2,\mathrm{3...}$

or

$\Delta x=|x_2-x_1|=(n+\frac{1}{2})\lambda n=0,1,2,...$

Electric Fields

$E=k\frac{q}{r^2}$, where $k=\frac{1}{4\pi {ϵ}_0}$

$k=9\times 10^9$

${ϵ}_0=8.85\times 10^{-12}$

Always point from positive charges to negative charges.

Mass of Proton: $1.672\times 10^{-27}$ kg

Mass of Electron: $9.109\times 10^{-31}$ kg

Elementary Charge: $1.6021\times 10^{-19}$ C

Electric Field of Line of Charge:

Finite: $E=k\frac{Q}{r(r^2+(\frac{L}{2}{)}^2{)}^{\frac{1}{2}}}$

Infinite: $E=\frac{\lambda }{2\pi {ϵ}_{0}r}$ or $k\frac{2|\lambda |}{r}$

$\lambda =\frac{C}{L}$

Electric Field of Disk of Charge

$\eta =\frac{Q}{\text{area}}=\frac{Q}{\pi R^2}$

Finite: $E=\frac{\eta }{2{ϵ}_0}(1-\frac{x}{\sqrt{x^2+R^2}})$, where R is radius of disk and x is distance from the disk.

Infinite (plane): $E=\frac{\eta }{2{ϵ}_0}$

Electric Field of Capacitor

$E_{cap}=\frac{\eta }{{ϵ}_0}$

Force of a capacitor plate on the other:

$F=E_{cap}\times q_{otherplate}$

Electric Field of Ring of Charge

$\vec{E}=k\frac{Qx}{(x^2+a^2{)}^{\frac{3}{2}}}$

Where

• x is the distance from the center of the ring
• a is the radius of the ring

Electric Field of a Dipole

${\vec{E}}_{dipole}=k\frac{2\vec{p}}{r^3}$ (On axis)

${\vec{E}}_{dipole}=-k\frac{\vec{p}}{r^3}$ (Bisecting plane)

Where: the direction of $\vec{p}$ identifies the orientation of the dipole

Sphere of Charge

${\vec{E}}_{sphere}=k\frac{Q}{r^2}$ for $r\le R$

Motion in a Uniform Field

$a=\frac{qE}{m}=\text{constant}$