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Question #91427

Difficulty: 5/10 | 🟢 Free

The Question

A particle of mass $m$ oscillates in a viscous medium that exerts a linear damping force $F_d = -bv$ , where $v$ is the velocity and $b$ is a small positive constant. The particle is subjected to a non-linear restoring force derived from the potential energy function $U(x) = \frac{1}{2}kx^2 + \frac{1}{4}\alpha x^4$ , where $k$ and $\alpha$ are positive constants. For a given cycle, the motion of the particle can be approximated as $x(t) \approx A \cos(\omega t)$ . The Quality Factor $Q$ of the oscillator is defined as $2\pi$ times the ratio of the maximum energy stored in the system to the energy dissipated per cycle. Which of the following expressions correctly represents the effective Quality Factor $Q$ as a function of the amplitude $A$ ?

Options

A.
$\frac{\sqrt{m} (k + \frac{1}{2}\alpha A^2)}{b \sqrt{k + \frac{3}{4}\alpha A^2}}$
B.
$\frac{\sqrt{m(k + \frac{3}{4}\alpha A^2)}}{b}$
C.
$\frac{\sqrt{m(k + \frac{1}{2}\alpha A^2)}}{b}$
D.
$\frac{\sqrt{m} (k + \frac{3}{4}\alpha A^2)}{b \sqrt{k + \frac{1}{2}\alpha A^2}}$

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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

Quality Factor (Q-factor)|Effective Spring Constant|Oscillations

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the fundamental definition of the Quality Factor against rote memorization of linear oscillator formulas.

Students often memorize the Q-factor for a linear oscillator as $Q = \frac{\sqrt{mk}}{b}$ (analogous to $\frac{1}{R}\sqrt{\frac{L}{C}}$ in LCR circuits).

When faced with a non-linear restoring force, the common trap is to find an 'effective spring constant' $k_{eff} = k + \frac{3}{4}\alpha A^2$ using harmonic balance, and then blindly substitute it into the linear $Q$ formula to get Option B.

This fundamentally violates the thermodynamic definition of $Q$ .

The stored energy in a non-linear potential is not $\frac{1}{2}k_{eff}A^2$ , but rather the exact integral of the force, $U(A) = \frac{1}{2}kA^2 + \frac{1}{4}\alpha A^4$ .

By forcing the use of the fundamental definition $Q = 2\pi \frac{E}{\Delta E}$ , this question exposes whether the student understands the energetic origin of the Q-factor or is merely pattern-matching linear differential equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Quality Factor (Q-factor)
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free