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Question #42351

Difficulty: 8/10 | 🟢 Free

The Question

A damped mechanical oscillator of mass $m$ , spring constant $k$ , and damping coefficient $b$ is driven by a force $F(t) = F_0 \cos(\omega t)$ . An analogous series LCR circuit with inductance $L=m$ , capacitance $C=1/k$ , and resistance $R=b$ is driven by a voltage $V(t) = F_0 \cos(\omega t)$ . Both systems operate in steady state at their resonant frequency $\omega_0 = \sqrt{k/m}$ . Let $E_{mech}$ be the total mechanical energy of the oscillator, and $U_{heat}$ be the heat dissipated in the LCR circuit over one complete cycle. Which of the following represents the ratio $\frac{E_{mech}}{U_{heat}}$ ?

Options

A.
$\frac{m \omega_0}{2\pi b}$
B.
1
C.
$\frac{2\pi m \omega_0}{b}$
D.
$\frac{b}{\pi m \omega_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Conservative vs Non-conservative|Driven Oscillations and Resonance|Damped Oscillation Equation|Transient Current in RL Circuit

Why This Question Is Tricky

Common Mistakes

This question targets a lethal misconception in driven dissipative systems: conflating steady-state resonance with a purely conservative state.

Because the total mechanical energy $E_{mech}$ remains strictly constant at resonance, students often incorrectly apply the Work-Energy Theorem to conclude that the net work done is zero and thus no energy is dissipated, or they assume the stored energy equals the dissipated energy (guessing a ratio of 1).

By mapping the mechanical system to an LCR circuit, the problem forces the realization that the driving source continuously supplies power to exactly match the non-conservative path-dependent heat loss ( $I^2R$ or $bv^2$ ) at every instant, while the conservative stored energy remains entirely decoupled from the per-cycle dissipation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Conservative vs Non-conservative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free