Question

LC osillations are similar and analogous to the mechanical oscillations of a block attached to a spring. The electrical equivalent of the force constant of the spring is

• A. reciprocal of capacitive reactance
• B. capacitive reactance
• C. reciprocal of capacitance
• D. capacitance

Answer 1

The Correct Option is C

Step 1: Recall the equation for mechanical oscillations of a block-spring system

The equation of motion for a simple harmonic oscillator (block-spring system) is given by:

$m \frac{d^2x}{dt^2} + kx = 0$

where:

$m$ is the mass of the block
$x$ is the displacement from equilibrium position
$k$ is the force constant of the spring

Step 2: Recall the equation for LC oscillations in an electrical circuit

For an ideal LC circuit (ignoring resistance), the voltage across the capacitor plus the voltage across the inductor must be zero, based on Kirchhoff's Voltage Law.

Voltage across capacitor: $V_C = \frac{Q}{C}$

Voltage across inductor (induced emf): $V_L = L \frac{dI}{dt} = L \frac{d}{dt} (\frac{dQ}{dt}) = L \frac{d^2Q}{dt^2}$

Setting the sum of voltages to zero (ignoring sign conventions for now for analogy):

$L \frac{d^2Q}{dt^2} + \frac{Q}{C} = 0$

Step 3: Compare the equations for mechanical and LC oscillations

Comparing the mechanical oscillation equation: $m \frac{d^2x}{dt^2} + kx = 0$
and the LC oscillation equation: $L \frac{d^2Q}{dt^2} + \frac{1}{C}Q = 0$

We can observe the following analogies between the physical quantities:

Displacement ($x$) in mechanical system is analogous to Charge ($Q$) in electrical system.
Mass ($m$) in mechanical system is analogous to Inductance ($L$) in electrical system.
Force constant ($k$) in mechanical system is analogous to $\frac{1}{C}$ in electrical system.

Step 4: Identify the electrical equivalent of the force constant

From the analogy in Step 3, the electrical equivalent of the force constant ($k$) of the spring is $\frac{1}{C}$, which is the reciprocal of capacitance.

Step 5: Check the options against the derived equivalent

Reciprocal of capacitive reactance: Capacitive reactance $X_C = \frac{1}{\omega C}$. Reciprocal is $\omega C$. This is frequency dependent and not a constant equivalent to $k$.

Capacitive reactance: $X_C = \frac{1}{\omega C}$. Frequency dependent, not a constant equivalent to $k$.

Reciprocal of capacitance: $\frac{1}{C}$. This matches our derived equivalent.

Capacitance: $C$. This is analogous to the reciprocal of the force constant (if we consider stiffness as reciprocal of force constant).

Step 6: Select the correct option

Based on the analogy derived, the electrical equivalent of the force constant of the spring is the reciprocal of capacitance.

Final Answer: The final answer is reciprocal of capacitance

Answer 2

LC oscillations are analogous to the mechanical oscillations of a block attached to a spring. In a mechanical spring-block system, the force constant ($k$) determines the restoring force. We want to find the electrical equivalent of the force constant in an LC circuit.

In a spring-block system, the angular frequency ($\omega$) of oscillation is given by $\omega = \sqrt{\frac{k}{m}}$, where $k$ is the force constant and $m$ is the mass. 

In an LC circuit, the angular frequency is given by $\omega = \frac{1}{\sqrt{LC}}$ 

Thus $ \sqrt{\frac{k}{m}} = \frac{1}{\sqrt{LC}} \implies \frac{k}{m} = \frac{1}{LC} $ 

Since k can be analogous to $\frac{1}{C}$, the electrical equivalent of the force constant is the reciprocal of capacitance.

Capacitive reactance ($X_C$) is given by $\frac{1}{\omega C}$. 

Its reciprocal is $\omega C$, which is not analogous to the force constant. The reciprocal of capacitance is $\frac{1}{C}$.

The correct answer is (C) reciprocal of capacitance.