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Question #42751

Difficulty: 8/10 | 🟢 Free

The Question

A thin rod of length $L$ is placed along the x-axis with its ends at $x = 0$ and $x = L$ . The rod has a non-uniform linear mass density given by $\lambda(x) = \alpha x$ , where $\alpha$ is a positive constant with units of $\text{kg/m}^2$ . What is the gravitational potential $V$ at a point $P$ located on the x-axis at $x = -L$ ? (Let $G$ be the universal gravitational constant).

Options

A.
$-G \alpha L (1 - \ln 2)$
B.
$-\frac{3}{10} G \alpha L$
C.
$-\frac{1}{2} G \alpha L \ln 2$
D.
$-G \alpha L \ln 2$

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

Superposition of Potentials|Potential vs Field Relationship|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the superposition principle for scalar fields in continuous, non-uniform mass distributions.

A common, fatal heuristic is treating an extended body as a point mass located at its center of mass ( $x_{cm} = 2L/3$ ), which incorrectly yields $V = -GM/r_{cm} = -3G\alpha L/10$ .

While the center of mass approximation is valid for calculating the external gravitational field of spherically symmetric bodies, it strictly fails for rods and non-symmetric distributions.

Students must recognize that potential requires the direct integration of $dV = -G dm/r$ , forcing them to account for the spatially varying density $\lambda(x)$ and the non-linear dependence of potential on distance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Superposition of Potentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free