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Question #43528

Difficulty: 8/10 | 🟢 Free

The Question

Two identical spherical molecules, each of mass $m$ and physical radius $R$ , are initially at rest at an infinite separation. They attract each other via a dipole-dipole interaction, but at short distances, the Pauli exclusion principle causes a strong core repulsion. The net effective potential energy of the two-molecule system is given by $U(r) = \frac{A}{r^6} - \frac{B}{r^3}$ , where $r$ is the center-to-center distance, and $A$ and $B$ are positive constants. Given that $\left(\frac{A}{B}\right)^{1/3} > 2R$ , what is the maximum relative momentum $p_{\text{rel}}$ achieved by the molecules, and what is their distance of closest approach $r_{\text{min}}$ ?

Options

A.
$p_{\text{rel}} = B \sqrt{\frac{m}{4A}}, \quad r_{\text{min}} = \left(\frac{A}{B}\right)^{\frac{1}{3}}$
B.
$p_{\text{rel}} = \sqrt{\frac{mB}{8R^3}}, \quad r_{\text{min}} = 2R$
C.
$p_{\text{rel}} = B \sqrt{\frac{m}{4A}}, \quad r_{\text{min}} = \left(\frac{2A}{B}\right)^{\frac{1}{3}}$
D.
$p_{\text{rel}} = B \sqrt{\frac{m}{2A}}, \quad r_{\text{min}} = \left(\frac{A}{B}\right)^{\frac{1}{3}}$

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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

Interatomic Force Curve|Dipole Potential Angular Dependence|Electric Field Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from macroscopic hard-sphere collision models to microscopic potential-driven interactions.

The primary trap lies in assuming that the collision ends at physical contact ( $r = 2R$ ), leading students to ignore the core-repulsion term ( $A/r^6$ ) and incorrectly calculate the maximum momentum and closest approach based purely on the attractive dipole potential (Option B).

Furthermore, it tests the correct application of reduced mass in two-body systems (Option D uses $\mu=m$ ) and distinguishes between the equilibrium separation where momentum is maximized and the classical turning point where the molecules reverse direction (Option C conflates the two).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Interatomic Force Curve
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free