A series LCR damped oscillatory circuit connected to a DC battery emf source can

Question

A series LCR damped oscillatory circuit connected to a DC battery emf source can be equated to spring-block oscillating system when the block is immersed in say some motor oil.( This topic is discussed in PHY 2170) Assuming an external force F sets the spring block system into damped oscillations write an equation F = _____and a similar equation for the damped electrical oscillator. From the following list of electrical quantities match appropriate quantities in the mechanical oscillating system to that on the electrical oscillating system V, R, L, C, q, I Mechanical Electrical Displacement (x) Force (F) Spring constant (k) Mass (m) Velocity (v) Acceleration (a) damping coefficient (gamma)

Explanation / Answer

The two differential equations are more or less identical:

For the 1D spring:

F = m d2x/dt2 = - b dx/dt (damping force) - k x (Hooke's law/restoring force).

Or d2x/dt2 + b/m dx/dt + k/m x = 0.

And for the RLC circuit:

0 = IR (Voltage drop over resistor) + q/C (Voltage drop over capacitor) + L dI/dt (Voltage drop across inductor).

Or d2q/dt2 + R/L dq/dt + 1/LC q = 0.

Do you see the similarities?

Both of these are non-driven oscillations. If you want to add driving, simple replace the 0 on the RHS with some F and V respectively.

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