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

Derivation from standard differential equation for SHM:

Springs

$w=\sqrt{\frac{k}{m}}$

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

$T=2\pi \sqrt{\frac{m}{k}}$

Pendulums

Simple

For $\theta <<1,\sin (\theta )\approx \theta$

Angular Frequency:

$w=2\pi f=\sqrt{\frac{g}{L}}$

Small amplitude: $ma=-mg\theta$

For small theta: $\theta ={\theta }_{max}\sin \sqrt{\frac{g}{L}}t$ (from hyperphysics)

Period can be approximated similarly to the spring: $T=2\pi \sqrt{\frac{L}{g}}$

Physical

$w=\sqrt{\frac{Mgl}{I}}$ (Small angle approx)

$T=2\pi \sqrt{\frac{I}{Mgl}}$

${\tau }_{net}=I\alpha$

$l$ = distance from pivot to center of mass

Damped Oscillations

$x(t)=A{e}^{-bt/2m}cos(wt+{\varphi }_0)$

$b=\text{damping constant}$

Where

$w=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}$

Energy: $E=E_0{e}^{\frac{-t}{\tau }}t=\tau \text{when }E=.37E_0$

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

Light

$c=f\lambda$

$c=299,792,458m/s$

Index of refraction $=\frac{\text{speed of light in vacuum (c)}}{\text{speed of light in the material (v)}}=\frac{c}{v}$

For material: $f_{mat}=f_{vac}$

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

Power

$I=\frac{P_{source}}{4\pi r^2}$

$I$ Is in $W/m^2$ and $P_{source}$ is in $W$

Decible

$\beta =10db{\log }_{10}(\frac{I}{I_0})$

$I_0=1\times 10^{-12}W/m^2$

Doppler Effect

For sound waves

Approaching Source

$f_{+}=\frac{f_0}{1-v_s/v}$ or $f_o=f_s(\frac{v_w}{v_w-v_s})$

Receding Source

$f_{-}=\frac{f_0}{1+v_s/v}$ or $f_o=f_s(\frac{v_w}{v_w+v_s})$

Approaching Observer

$f_{+}=(\frac{1+v_o}{v})f_0$ or $f_o=f_s(\frac{v_w+v_o}{v_w})$

Receding Observer

$f_{-}=(\frac{1-v_o}{v})f_0$ or $f_o=f_s(\frac{v_w-v_o}{v_w})$

For observers the wavelength doesn't change when they move.

Bouncing sound (object approaching):

$f_o=f_i(\frac{v_s+v_{obj}}{v_s})(\frac{v_s}{v_s-v_{obj}})$

For light waves

Receding Source

${\lambda }_{-}=\sqrt{\frac{1+v_s/c}{1-v_s/c}}{\lambda }_0$

Approaching Source

${\lambda }_{+}=\sqrt{\frac{1-v_s/c}{1+v_s/c}}{\lambda }_0$

Superposition

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

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

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:

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

Maximum destructive:

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