Animated Solution for Physics - Oscillations: An $L$-$C$-$R$ circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant $b$, the correct equivalence would be

Finding Connections

Physics is not just a collection of isolated formulas; it is a beautifully interconnected web of ideas. One of the most profound connections in classical physics is the electro-mechanical analogy.

Imagine you are looking at two completely different systems. On one side, you have an electrical circuit with an inductor, a capacitor, and a resistor connected to an alternating voltage source. On the other side, you have a mechanical system consisting of a block of mass attached to a spring, moving through a viscous fluid that provides damping, while being pushed by an external oscillating force.

At first glance, these systems seem to have nothing in common. One deals with invisible electrons flowing through wires, while the other deals with tangible masses and springs. However, as we will see, the underlying mathematics governing their behavior is exactly the same!

Let's start by analyzing the electrical $L$-$C$-$R$ circuit. We can apply Kirchhoff's Voltage Law (KVL), which states that the sum of the voltage drops across all components in a closed loop must equal the applied electromotive force (EMF).

The voltage drop across the inductor is $L\frac{di}{dt}$, across the resistor is $iR$, and across the capacitor is $\frac{q}{C}$. Equating this to the applied AC voltage $E=E_0\sin \omega t$, we get:

To make this equation more fundamental, we need to express it in terms of a single variable. We know that current $i$ is simply the rate of flow of charge, so $i=\frac{dq}{dt}$. Consequently, the rate of change of current is the second derivative of charge, $\frac{di}{dt}=\frac{d^{2}q}{d{t}^2}$.

Substituting these into our KVL equation, we obtain a second-order linear differential equation:

Now, let's shift our focus to the mechanical spring-mass-damper system. We will use Newton's Second Law of Motion, which states that the net force acting on an object is equal to its mass times its acceleration ($F_{\text{net}}=ma$).

There are three forces acting on the mass: 1. The restoring force from the spring, which is $-kx$ (Hooke's Law). 2. The damping force from the fluid, which opposes motion and is proportional to velocity, $-bv$. 3. The external driving force, $F_0\cos \omega t$.

Setting the sum of these forces equal to $ma$, we get:

Just like we did for the electrical circuit, let's express velocity and acceleration in terms of displacement $x$. Velocity is $v=\frac{dx}{dt}$, and acceleration is $a=\frac{d^{2}x}{d{t}^2}$.

Rearranging the terms to bring all the derivatives to one side, we get another second-order linear differential equation:

Comparing the Equations

This is where the magic happens. Let's place our two differential equations side by side and observe their structure.

Electrical:

Mechanical:

Notice how perfectly they mirror each other! By comparing the coefficients of the corresponding terms, we can establish a direct dictionary between electrical and mechanical quantities:

1. Charge ($q$) $\leftrightarrow$ Displacement ($x$): Both represent the fundamental state variable of the system. 2. Inductance ($L$) $\leftrightarrow$ Mass ($m$): Both provide inertia. Mass resists changes in velocity, while inductance resists changes in current. 3. Resistance ($R$) $\leftrightarrow$ Damping Constant ($b$): Both are responsible for the dissipation of energy (as heat) from the system. 4. Inverse Capacitance ($\frac{1}{C}$) $\leftrightarrow$ Spring Constant ($k$): A stiff spring (large $k$) is analogous to a small capacitor (large $\frac{1}{C}$), both requiring a large "force" (mechanical force or electrical voltage) to produce a small "displacement" (distance or charge).

Therefore, the correct equivalence is $L\leftrightarrow m$, $C\leftrightarrow \frac{1}{k}$, and $R\leftrightarrow b$. This profound analogy allows engineers to simulate complex mechanical vibrations using easily adjustable electrical circuits!