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Question #42265

Difficulty: 8/10 | 🟢 Free

The Question

Two identical uniform solid disks, each of mass $M$ and radius $R$ , are placed parallel to the $xy$ -plane such that their centers lie on the $z$ -axis. The first disk is centered at the origin $(0,0,0)$ and the second disk is centered at $(0,0,3R)$ . A student is tasked with finding the absolute gravitational potential at the point $P(0,0,4R)$ on the $z$ -axis, with the standard reference $V \to 0$ as $r \to \infty$ . Which of the following represents the correct potential at $P$ ?

Options

A.
$\frac{2GM}{R} (5 - \sqrt{17} - \sqrt{2})$
B.
$\frac{2GM}{R} (7 - \sqrt{17} - \sqrt{2})$
C.
$-\frac{2GM}{R} (\sqrt{17} + \sqrt{2})$
D.
$-\frac{5GM}{4R}$

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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

Gravitational Potential Reference|Principle of Superposition|Ampere’s Circuital Law Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly establish the absolute gravitational potential for non-spherical bodies by integrating the gravitational field or mass elements, while strictly enforcing the boundary condition $V \to 0$ as $r \to \infty$ .

A common lethal mistake (Trap B) occurs when students integrate the gravitational field $g(z)$ starting from the center of the disk and implicitly set the constant of integration such that $V(0) = 0$ .

Because the bodies are non-spherical, the potential difference $\Delta V$ from the center to a point $z$ does not represent the absolute potential.

Failing to unify the reference point at infinity for both overlapping fields results in an additive error of $+2GM/R$ per disk.

Other traps include incorrectly evaluating the lower limit of the radial integration (Trap C) or improperly treating the extended bodies as point masses (Trap D).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Potential Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free