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Question #43546

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is constrained to move along the positive x-axis. Its potential energy is given by $U(x) = U_0 \left[ \frac{1}{3} \left(\frac{a}{x}\right)^3 - \frac{1}{2} \left(\frac{a}{x}\right)^2 \right]$ where $U_0$ and $a$ are positive constants. The particle is initially at rest at its stable equilibrium position. A short impulse $J$ is delivered to the particle in the positive x-direction. Given that $J \ll \sqrt{m U_0}$ , the maximum x-coordinate reached by the particle is approximately:

Options

A.
$a + \frac{J a}{\sqrt{m U_0}}$
B.
$a + \frac{J a}{\sqrt{2 m U_0}}$
C.
$a + \frac{2 J a}{\sqrt{m U_0}}$
D.
$\frac{J a}{\sqrt{m U_0}}$

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

SHM Definition via Differential Equation|Interatomic Force Curve|Stress vs. Strain Definition

Why This Question Is Tricky

Common Mistakes

This question tests the ability to model small oscillations in an arbitrary non-linear potential well, a core JEE Advanced concept.

The primary trap lies in the miscalculation of the effective spring constant $k_{eff}$ .

Students often confuse the Taylor expansion term $\frac{1}{2}U''(x_0)(\Delta x)^2$ with $U''(x_0)(\Delta x)^2$ , leading to a missing factor of $1/2$ in the energy equation and incorrectly selecting Option B.

Additionally, Option D traps students who calculate the amplitude correctly but forget to add the equilibrium offset $x_0$ .

The problem forces the use of the Work-Energy theorem in a localized quadratic approximation, as the exact potential cannot be solved algebraically for $x_{max}$ without it.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	SHM Definition via Differential Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free