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Question #43516

Difficulty: 8/10 | 🟢 Free

The Question

A satellite is in a circular orbit of radius $R$ around a planet of mass $M$ . As part of its structure, it contains a uniform straight rod of mass $m$ , length $L$ ( $L \ll R$ ), cross-sectional area $A$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ . The rod is gravity-gradient stabilized such that it always aligns along the radial direction towards the center of the planet. The rod is mounted inside a rigid, massless, non-expanding frame that holds its ends, strictly constraining its total length to remain $L$ . If the temperature of the rod increases by $\Delta T$ , what is the magnitude of the compressive force $F_c$ exerted by the frame on each end of the rod? (Assume the orbital radius $R$ remains constant and $G$ is the universal gravitational constant).

Options

A.
$AY \alpha \Delta T + \frac{GMmL}{4R^3}$
B.
$AY \alpha \Delta T - \frac{GMmL}{4R^3}$
C.
$AY \alpha \Delta T + \frac{3GMmL}{8R^3}$
D.
$AY \alpha \Delta T + \frac{GMmL}{2R^3}$

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

Geostationary Satellite Constraints|Thermal Stress in Constrained Rods|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of orbital tidal forces (gravity gradient) and thermo-elasticity.

A common trap is to either ignore the tidal tension entirely or to incorrectly assume it compresses the rod (leading to a sign error).

Furthermore, students often fail to integrate the varying tidal tension $T(x)$ over the rod's length, mistakenly using the maximum tension at the center ( $3m\omega^2 L / 8$ ) as a uniform value, which leads to the incorrect coefficient in the tidal term.

The correct approach requires superimposing the mechanical strain from the distributed tidal force with the uniform thermal strain, and applying the zero-net-extension boundary condition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Geostationary Satellite Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free