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Question #42311

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ moves in one dimension $x > 0$ under the influence of a conservative force field. The potential energy of the particle is given by $U(x) = U_0 \left[ \ln\left(\frac{x}{a}\right) + \cos\left(\frac{x}{a}\right) \right]$ where $U_0$ and $a$ are positive constants. For large values of $x$ ( $x \gg a$ ), the positions of the stable equilibrium points are approximately given by (where $n$ is a large positive integer):

Options

A.
$x \approx 2n\pi a$
B.
$x \approx \left(2n + \frac{1}{2}\right)\pi a$
C.
$x \approx (2n + 1)\pi a$
D.
$x \approx \left(2n + \frac{3}{2}\right)\pi a$

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

Variable Force Integration|Equilibrium Stability|Graphical Concavity Significance

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-linear, transcendental potential energy surfaces.

A common trap is assuming that setting $F=0$ and finding the asymptotic roots of $\sin(x/a) = 0$ is sufficient, which leaves students guessing between $2n\pi$ and $(2n+1)\pi$ .

Students often fail to apply the second derivative test ( $d^2U/dx^2 > 0$ ) to distinguish between stable and unstable equilibria.

Furthermore, the interaction between the decaying logarithmic potential and the oscillatory harmonic potential requires asymptotic reasoning ( $x \gg a$ ) to isolate the dominant term in the concavity condition, effectively filtering out candidates who rely purely on algebraic manipulation without physical intuition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Variable Force Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free