Before You Begin

Question #43151

Difficulty: 8/10 | 🟢 Free

The Question

An LCR series circuit contains an inductor $L$ , a capacitor $C$ , and a non-linear resistor across which the voltage drop is given by $V_R = \beta i^3$ , where $i$ is the current and $\beta$ is a positive constant. The circuit is driven by an AC voltage source $V(t) = V_0 \cos(\omega_0 t)$ , where $\omega_0 = 1/\sqrt{LC}$ is the resonant frequency. Assume that the steady-state current can be approximated as purely sinusoidal at the fundamental frequency $\omega_0$ .

A separate mechanical damped harmonic oscillator of mass $m$ and spring constant $k$ is subjected to a linear damping force $F_d = -b v$ , where $v$ is the velocity and $b$ is the damping constant. It is driven by an external force $F(t) = F_0 \cos(\omega_0 t)$ at its resonant frequency $\omega_0 = \sqrt{k/m}$ .

If the average power dissipated over one complete cycle in both systems is equal, which of the following expressions gives the equivalent linear damping constant $b$ ?

Options

A.
$\left( \frac{3 \beta F_0^6}{4 V_0^4} \right)^{1/3}$
B.
$\left( \frac{\beta F_0^6}{V_0^4} \right)^{1/3}$
C.
$\left( \frac{4 \beta F_0^6}{3 V_0^4} \right)^{1/3}$
D.
$\left( \frac{\beta F_0^6}{2 V_0^4} \right)^{1/3}$

Check Your Answer

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

Average Power over Cycle|Driven Oscillations and Resonance|Calculus: Definite Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to evaluate average power dissipation in driven non-linear oscillators by integrating over a full period, rather than incorrectly equating instantaneous peak values.

A common trap is to assume that the peak voltage equals the peak resistive drop ( $V_0 = \beta I_0^3$ ), which ignores the generation of higher harmonics and violates energy balance over a cycle.

By requiring the student to equate the average power supplied ( $\langle V i \rangle$ ) to the average power dissipated ( $\langle \beta i^4 \rangle$ ), the problem enforces the use of the first harmonic approximation.

Furthermore, it tests the exact mapping of energy dissipation between an electrical LCR circuit and a mechanical damped oscillator, ensuring dimensional and physical consistency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Average Power over Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free