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Question #43682

Difficulty: 8/10 | 🟢 Free

The Question

A sealed capsule of volume $V$ contains $N$ nuclei of a specific isotope. The total mass of the empty capsule plus the sum of the rest masses of all individual constituent nucleons of these $N$ nuclei is $M_0$ . The capsule is fully submerged in an incompressible liquid of density $\rho$ inside a tank. The tank undergoes a constant horizontal acceleration $a = \sqrt{3}g$ , where $g$ is the acceleration due to gravity. An external agent moves the capsule quasi-statically through a distance $L$ exactly along the direction of the effective gravitational field (deeper into the liquid). If the work done by the external agent is $W$ , which of the following expressions correctly represents the total binding energy $E_B$ of the $N$ nuclei? (Let $c$ be the speed of light in vacuum, and assume $\rho V > M_0$ ).

Options

A.
$\left( M_0 - \rho V + \frac{W}{2gL} \right) c^2$
B.
$\left( \rho V - M_0 - \frac{W}{2gL} \right) c^2$
C.
$\left( M_0 - \rho V + \frac{W}{gL} \right) c^2$
D.
$\left( M_0 - \rho V + \frac{W}{\sqrt{3}gL} \right) c^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Mass Defect vs. Binding Energy|Accelerated Frame Hydrostatics|Energy Density Locality

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of mass-energy equivalence and fluid statics in non-inertial frames.

The primary trap lies in the student's potential failure to recognize that the binding energy manifests as a macroscopic mass defect ( $\Delta M$ ) which directly alters the gravitational and inertial mass of the system, thereby affecting the macroscopic work done against buoyancy.

Additionally, students may misapply the Equivalence Principle by using standard gravity $g$ or the horizontal acceleration $a$ instead of the effective gravity $g_{eff} = \sqrt{g^2 + a^2}$ for the hydrostatic pressure gradient.

Finally, a common sign error occurs when students incorrectly add the mass defect to the constituent mass instead of subtracting it, leading to an inverted expression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Mass Defect vs. Binding Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free