# Fundamental Physics/Motion/Oscillation

## Oscillation

Motion of an object about an equilibrium position repeates itself over a period of time

## Spring's oscillation

Movement up and down of spring over a period of time $F_{a}=-F_{y}$ $ma=-ky$ $a=-{\frac {k}{m}}y$ ${\frac {d^{2}}{dt^{2}}}y=-{\frac {k}{m}}y$ $y=A\sin \omega t$ $\omega ={\sqrt {\frac {k}{m}}}=\lambda f$ Movement side by side of spring over a period of time

$F_{a}=-F_{x}$ $ma=-kx$ $a=-{\frac {k}{m}}x$ ${\frac {d^{2}}{dt^{2}}}x=-{\frac {k}{m}}x$ $x=A\sin \omega t$ $\omega ={\sqrt {\frac {k}{m}}}=\lambda f$ ## Pendulum oscillation

Movement side by side swinging of pendulum over a period of time $y^{''}(t)=-\omega y(t)$ $y(t)=A\sin \omega t$ $\omega ={\sqrt {\frac {l}{g}}}$ ## Electrical oscillation

### Series LC When circuit is at equilibrium, the total voltages are zero

$V_{L}+V_{C}=0$ $L{\frac {d}{dt}}i+{\frac {1}{C}}\int i=0$ ${\frac {d^{2}}{dt^{2}}}i+{\frac {1}{LC}}i=0$ ${\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i$ $i=A\sin \omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$  When circuit is at resonance, the total impedances are zero and the total voltages are zero

$Z_{L}+Z_{C}=0$ $V_{L}+V_{C}=0$ $V(\theta )=A\sin(\omega _{o}+2\pi )-A\sin(\omega _{o}-2\pi )$ $\omega _{o}=\pm j{\sqrt {\frac {1}{T}}}$ $T=LC$  ### Series RLC When circuit is at equilibrium, the total voltages are zero

$V_{L}+V_{C}+V_{R}=0$ $L{\frac {d}{dt}}i+{\frac {1}{C}}\int i+iR=0$ ${\frac {d^{2}}{dt^{2}}}i+{\frac {R}{L}}{\frac {d}{dt}}i+{\frac {1}{LC}}i=0$ ${\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i$ $i=A(\alpha )\sin \omega t$ $\alpha =Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$ $\beta ={\sqrt {\frac {1}{T}}}={\frac {1}{LC}}$ $\gamma =\beta \gamma ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ When R is shorted, $R=0$ and $\alpha =0$ ${\frac {d^{2}}{dt^{2}}}i=-\beta i$ $i=A\sin(\omega t)$ $\omega ={\sqrt {\beta }}$ $\beta ={\sqrt {\frac {1}{T}}}={\frac {1}{LC}}$ ## Electromagnetic oscillation $\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$ $E=A\sin \omega t$ $B=A\sin \omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=\mu \epsilon$  