Concept:
Mathematical analogies allow engineers to model electrical networks and mechanical systems using identical differential equations. Under the Current-Force ($I \rightarrow F$) Analogy (often extended to the Current-Torque $I \rightarrow T$ analogy for rotational systems), electrical nodes and currents correspond to mechanical junctions and torques/forces respectively. The dual relationships map across the foundational variables.
Step 1: Map the fundamental parameters under Current-Torque (or Current-Force) analogy.
In a parallel RLC network analyzed via nodal analysis, Kirchhoff's Current Law yields:
\[
I(t) = C \frac{dv}{dt} + \frac{1}{L}\int v \, dt + \frac{v}{R}
\]
For a rotational mechanical system, the torque balance equation is:
\[
T(t) = J \frac{d\omega}{dt} + K \int \omega \, dt + B\omega
\]
Equating matching algebraic terms establishes the standard variable pairings:
* Current ($I$) maps directly to Torque ($T$).
* Voltage ($v$ or $A$) maps directly to Angular Velocity ($\omega$ or IV).
Step 2: Map the accumulation variables.
* Flux Linkages ($\lambda$ or B): By definition, voltage is the derivative of flux linkages ($v = \frac{d\lambda}{dt} \implies \lambda = \int v \, dt$). In the mechanical system, the integral of angular velocity is angular displacement ($\theta = \int \omega \, dt$). Therefore, flux linkages map directly to displacement (I).
* Capacitance ($C$ or C): The coefficient matching the first derivative of the primary state variable ($\frac{dv}{dt}$ matching $\frac{d\omega}{dt}$) means that electrical capacitance ($C$) maps directly to the rotational Moment of Inertia ($J$ or II).
Step 3: Combine findings to verify options.
* A $\rightarrow$ IV (Voltage maps to Angular Velocity)
* B $\rightarrow$ I (Flux Linkages map to Displacement)
* C $\rightarrow$ II (Capacitance maps to Moment of Inertia)
This structural mapping matches Option (A).
