Analogy between LC circuit and mass–spring system.

The idea

An LC circuit and a mass–spring system both oscillate by exchanging energy between two stores. In the mechanical system, energy shifts between the kinetic store of the moving mass and the elastic potential store of the spring. In the electrical system, energy shifts between the magnetic field of the inductor and the electric field of the capacitor. Mass corresponds to inductance, as both resist changes in motion or current. The spring constant corresponds to the reciprocal of capacitance. The analogy helps derive the resonant angular frequency ω = 1/√(LC) by comparing it with ω = √(k/m). Damping by friction or resistance reduces the amplitude of oscillation in both systems. Voltage maps to force, charge to displacement, and current to velocity.

What earns the marks

• States that both systems exchange energy between two stores and oscillate at a natural frequency.
• Matches the mass to the inductance, because mass opposes changes in velocity and inductance opposes changes in current.
• Matches the spring to the capacitor, because the spring stores elastic potential energy and the capacitor stores electric field energy.
• Compares the resonant angular frequency of the LC circuit, ω = 1/√(LC), with that of the mass–spring system, ω = √(k/m).
• Explains that increasing inductance or mass lowers the natural frequency, while increasing the reciprocal of capacitance or the spring constant raises it.
• Describes damping in the mechanical system as friction or air resistance and in the electrical system as resistance, both reducing oscillation amplitude.

Common mistakes and how to put them right

• Matching capacitance directly to the spring constant.

  Put it right by matching capacitance to the reciprocal of the spring constant, since ω = 1/√(LC) and ω = √(k/m).

• Claiming that the capacitor stores magnetic energy.

  Put it right by stating that the capacitor stores energy in its electric field and the inductor stores energy in its magnetic field.

• Ignoring resistance or friction when describing ideal oscillations.

  Put it right by noting that resistance in the LC circuit and friction in the mass–spring system cause damping.

• Confusing the roles of current, charge and voltage with velocity, displacement and force.

  Put it right by treating current as analogous to velocity, charge as analogous to displacement, and voltage as analogous to force.

Exam tips

• Draw the two systems side by side and label the corresponding quantities before writing your explanation.
• Use the equations ω = 1/√(LC) and ω = √(k/m) to justify each part of the analogy rather than relying on memory alone.
• When discussing energy, name the electric field store, magnetic field store, kinetic store and elastic potential store explicitly.
• Check that any comparison you make is consistent with the equations, especially the inverse relationship between capacitance and frequency.