Motion of a complete circle

Distance	${\displaystyle s}$	${\displaystyle 2\pi r}$
Time	${\displaystyle t}$	${\displaystyle t}$
Speed	${\displaystyle v}$	${\displaystyle {\frac {2\pi }{t}}r=\omega r}$
Angular speed	${\displaystyle \omega }$	${\displaystyle {\frac {2\pi }{t}}=2\pi f}$
Frequency	${\displaystyle f}$	${\displaystyle {\frac {1}{t}}}$

Motion of an arc of a circle

Distance	${\displaystyle s}$	${\displaystyle \theta r}$
Time	${\displaystyle t}$	${\displaystyle t}$
Speed	${\displaystyle v}$	${\displaystyle {\frac {\theta }{t}}r}$
Angular speed	${\displaystyle \omega }$	${\displaystyle {\frac {\theta }{t}}=\theta f}$
Frequency	${\displaystyle f}$	${\displaystyle {\frac {1}{t}}}$

Spring's oscillation

Pendulum oscillation

At equilibrium with resistance not equal to zero

${\displaystyle V_{L}+V_{C}+V_{R}=0}$

${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0}$

${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {R}{2L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0}$

${\displaystyle {\frac {d^{2}i}{dt^{2}}}=-2\alpha {\frac {di}{dt}}-\beta i}$

${\displaystyle i=A(\alpha )Sin\omega t}$

${\displaystyle A(\alpha )=Ae^{\alpha t}t}$

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$

${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$

${\displaystyle T=LC}$

${\displaystyle \alpha =\beta \gamma }$

${\displaystyle \gamma =RC}$

At resonance with resistance not equal to zero

${\displaystyle Z_{L}+Z_{C}=0}$

${\displaystyle Z_{t}=R}$

${\displaystyle i(\omega =0)=0}$

${\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}$

${\displaystyle i(\omega =00)=0}$

At equilibrium with resistance equals zero

${\displaystyle V_{L}+V_{C}=0}$

${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt=0}$

${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {1}{LC}}i=0}$

${\displaystyle {\frac {d^{2}i}{dt^{2}}}=-{\frac {1}{T}}i}$

${\displaystyle i=ASin\omega t}$

${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$

${\displaystyle T=LC}$

At resonance with resistance not equal to zero

${\displaystyle Z_{L}+Z_{C}=0}$

${\displaystyle V_{L}+V_{C}=0}$

${\displaystyle V(\theta )=ASin(\omega _{o}t+2\pi )+ASin(\omega _{o}t-2\pi )}$