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Question #42255

Difficulty: 8/10 | 🟢 Free

The Question

Two heavy point masses, each of mass $M$ , are fixed at coordinates $(d, 0, d)$ and $(-d, 0, d)$ in a standard $xyz$ coordinate system where the z-axis is vertical. A small block of mass $m$ moves along the y-axis on a rough horizontal table lying in the $xy$ -plane ( $z=0$ ). The coefficient of kinetic friction between the block and the table is $\mu$ . The block is moved from $A(0, d, 0)$ to the origin $O(0, 0, 0)$ . Earth's uniform gravitational field $g$ acts in the $-z$ direction. Assuming the block never loses contact with the table, what is the sum of the work done on the block by the gravitational field of the two masses and the force of friction?

Options

A.
$\frac{GMm}{d} \left( \sqrt{2} - \frac{2 - \mu}{\sqrt{3}} \right) - \mu mg d$
B.
$\frac{GMm}{d} \left( \sqrt{2} - \frac{2}{\sqrt{3}} \right) - \mu mg d$
C.
$\frac{GMm}{d} \left( \sqrt{2} - \frac{2 + \mu}{\sqrt{3}} \right) - \mu mg d$
D.
$\frac{GMm}{d} \left( 2 - \sqrt{2} + \frac{\mu}{\sqrt{3}} \right) - \mu mg d$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Principle of Superposition|Conservative Field Path Independence|Work Done by Friction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize the principle of superposition for conservative fields with the path-dependent nature of non-conservative forces.

A common trap is assuming the normal force is simply $mg$ , ignoring the vertical component of the 3D gravitational field which makes the friction force variable and requires a line integral.

Students must recognize that while the work done by gravity is path-independent and easily found via scalar potential differences, the work done by friction demands explicit integration of the position-dependent normal force.

Furthermore, it tests spatial visualization to correctly assign signs, as the masses pull the block upwards, reducing the normal force rather than increasing it.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Principle of Superposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free