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# Kirchoff's Laws : Mass-Spring Equation

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Consider a basic electric circuit with a resistor, capacitor, and inductor and input voltage V(t). It follows Kirchoff's Laws that the charge on the capacitor Q = Q(t) solves the differential equation: {see attachment}, where L (inductance), R (resistance), and C (capacitance) are positive constants (depending on material). The current in the circuit is the rate of change of the charge: I(t) = Q'(t).

(a) Compare this to the mass-spring equation. What plays the role of the mass, spring constant, damping coefficient, displacement, velocity, external forcing?
(b) Suppose a given circuit has no input voltage (V(t)=0) and a positive initial charge Q(0) = Qo > 0. Find a condition on R > 0 so that the equation has oscillatory solutions.
(c) If you wanted to build a circuit that would oscillate forever without any input voltage (V(t)=0) what could you do?

**Please see attachment for diagram. Note: no computer, no calculator. Show how you would have done things by hand. Thanks very much!

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ODEs and Kirchoff's Law are investigated. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.

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(a)
The mass-spring equation is where m is mass; b is damping constant; k is spring force constant; x is displacement of mass from its original position; f(t) is applied external force.
The ...

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