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Question #42287

Difficulty: 8/10 | 🟢 Free

The Question

Two identical massive spheres, each of mass $M$ , are fixed at positions $(d, 0, 0)$ and $(-d, 0, 0)$ in free space. A small rigid probe of length $L$ ( $L \ll d$ ) moves along a circular equipotential path of radius $R=d$ in the $y-z$ plane, centered at the origin. The probe maintains its orientation parallel to the $x$ -axis throughout its motion. Although the probe's center of mass is in free fall, a tidal force acts across it. What is the magnitude of the relative tidal acceleration between the two ends of the probe?

Options

A.
$\frac{GM L}{2\sqrt{2} d^3}$
B.
$\frac{GM L}{\sqrt{2} d^3}$
C.
$\frac{3GM L}{4\sqrt{2} d^3}$
D.
$\frac{GM L}{4\sqrt{2} d^3}$

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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

Weightlessness in Orbit|Equipotential Surface Geometry|Superposition Principle

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that an object in free-fall along an equipotential surface experiences true weightlessness (zero internal stress).

Students often equate an 'equipotential path' with a 'zero-force environment' or assume the point-mass approximation applies perfectly to extended bodies.

By forcing the calculation of the second-order spatial derivative of the potential (the tidal tensor component), the problem distinguishes between first-order apparent weightlessness (center of mass acceleration) and the second-order geodesic deviation (tidal stretching) across the rigid body.

It traps students who only calculate the radial gradient or fail to apply the superposition principle to the field's curvature.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Weightlessness in Orbit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free