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Question #43922

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The Question

A mechanical oscillator (mass $M = 3 \text{ kg}$ , spring constant $K = 100 \text{ N/m}$ ) is modeled electrically using the Force-Voltage analogy, where velocity maps to current. The resulting equivalent ideal LC tank circuit is connected in series with a physical RL coil having inductance $L_0 = 1 \text{ H}$ and resistance $R_0 = 8 \text{ }\Omega$ .

At $t = 0$ , with the circuit initially relaxed (zero current, zero charge), an impulsive voltage $V(t)$ is applied across the entire series combination such that $\int_{0^-}^{0^+} V(t) dt = 24 \text{ V}\cdot\text{s}$ . Which of the following expressions gives the charge $q(t)$ on the capacitor for $t > 0$ ?

Options

A.
$2 e^{-4t} \sin(3t) \text{ C}$
B.
$\frac{\sqrt{6}}{2} e^{-t} \sin(2\sqrt{6} t) \text{ C}$
C.
$1.2 e^{-t} \sin(5t) \text{ C}$
D.
$2\sqrt{6} e^{-t} \sin(2\sqrt{6} t) \text{ C}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Effective Spring Constant|Transient Current in RL Circuit|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electromechanical analogies with transient circuit dynamics.

The primary trap lies in compartmentalizing the circuit components.

Students often calculate the damping factor $\gamma = R/2L$ using only the physical inductor $L_0 = 1 \text{ H}$ , yielding an incorrect $\gamma = 4 \text{ s}^{-1}$ and a deceptively clean damped frequency of $3 \text{ rad/s}$ (Option A).

They fail to recognize that in a series loop, the 'mechanical' inductance $L_m$ (representing mass) contributes to the total inertia of the system, meaning the RL decay envelope is governed by the total equivalent inductance $L_{eq} = L_m + L_0$ .

Secondary traps include applying the impulse only to $L_0$ (Option D) or using the undamped natural frequency for the amplitude calculation (Option C).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Effective Spring Constant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free