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{\bf Formulas}

Gravitational force
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near earth's surface:  $mg$
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far from earth's surface:  ${mgR^2 \over (R + x)^2}$ where $R$ is the radius of the earth.

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Definition:  The Wronskian of two differential functions, $f$ and $g$ is

\centerline{$W(f, g) = fg' - f'g = \big|\matrix{f & g \cr f' & g'}\big|$}

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$cos (y \mp x) = cos(x \mp y) = cos(x) cos(y) \pm sin(x)sin(y)$

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Mechanical Vibrations:

\centerline{$mu''(t) + \gamma u'(t) + ku(t) = F_{external}, ~~m, \gamma, k \geq 0$} 

\centerline{ $mg - kL = 0$, ~~~~~$F_{viscous}(t) = \gamma u'(t)$}

$m$=  mass,
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$k$ = spring force proportionality constant,
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$\gamma$ = damping force proportionality constant

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g = 9.8 m/sec


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Electrical Vibrations:
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\centerline{$L{dI(t) \over dt} + RI(t) + {1 \over C}Q(t) = E(t), ~~L, R, C \geq 0 {\hbox{ and 
}}
I = {dQ \over dt}$~~~~~~}

$L$ = inductance (henrys),
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$R$ = resistance (ohms)
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$C$ = capacitance (farads)
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$Q(t)$ = charge at time $t$ (coulombs)
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$I(t)$ = current at time $t$ (amperes)
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$E(t)$ = impressed voltage (volts).

1 volt = 1 ohm $\cdot$ 1 ampere = 1 coulomb / 1 farad = 1 henry $\cdot$ 1 amperes/ 1 second

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