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

${\displaystyle F_{a}=-F_{y}}$
${\displaystyle ma=-ky}$
${\displaystyle a=-{\frac {k}{m}}y}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}y=-{\frac {k}{m}}y}$
${\displaystyle y=A\sin \omega t}$
${\displaystyle \omega ={\sqrt {\frac {k}{m}}}=\lambda f}$

Movement side by side of spring over a period of time

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

Pendulum oscillation

Movement side by side swinging of pendulum over a period of time

${\displaystyle y^{''}(t)=-\omega y(t)}$
${\displaystyle y(t)=A\sin \omega t}$
${\displaystyle \omega ={\sqrt {\frac {l}{g}}}}$

Electrical oscillation

Series LC

When circuit is at equilibrium, the total voltages are zero

${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int i=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i+{\frac {1}{LC}}i=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i}$
${\displaystyle i=A\sin \omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$

When circuit is at resonance, the total impedances are zero and the total voltages are zero

${\displaystyle Z_{L}+Z_{C}=0}$
${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle V(\theta )=A\sin(\omega _{o}+2\pi )-A\sin(\omega _{o}-2\pi )}$
${\displaystyle \omega _{o}=\pm j{\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$

Series RLC

When circuit is at equilibrium, the total voltages are zero

${\displaystyle V_{L}+V_{C}+V_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int i+iR=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i+{\frac {R}{L}}{\frac {d}{dt}}i+{\frac {1}{LC}}i=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i}$
${\displaystyle i=A(\alpha )\sin \omega t}$
${\displaystyle \alpha =Ae^{-\alpha t}}$
${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$
${\displaystyle \beta ={\sqrt {\frac {1}{T}}}={\frac {1}{LC}}}$
${\displaystyle \gamma =\beta \gamma ={\frac {R}{2L}}}$
${\displaystyle T=LC}$
${\displaystyle \gamma =RC}$

When R is shorted, ${\displaystyle R=0}$ and ${\displaystyle \alpha =0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-\beta i}$
${\displaystyle i=A\sin(\omega t)}$
${\displaystyle \omega ={\sqrt {\beta }}}$
${\displaystyle \beta ={\sqrt {\frac {1}{T}}}={\frac {1}{LC}}}$

Electromagnetic oscillation

${\displaystyle \nabla ^{2}E=-\omega E}$
${\displaystyle \nabla ^{2}B=-\omega B}$
${\displaystyle E=A\sin \omega t}$
${\displaystyle B=A\sin \omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=\mu \epsilon }$