Before You Begin

Question #43678

Difficulty: 8/10 | 🟢 Free

The Question

In the semi-empirical Liquid Drop Model, atomic nuclei are treated as incompressible spherical droplets. Consider two nuclei, Aluminum ( $^{27}\text{Al}$ ) and Tellurium ( $^{125}\text{Te}$ ), completely submerged in a hypothetical, non-viscous 'pion fluid' of uniform density $\rho_\pi = 1.0 \times 10^{17} \text{ kg/m}^3$ . Both nuclei are released from rest and subjected to a uniform strong-field acceleration $g_s = 10^{20} \text{ m/s}^2$ . Assume the nuclear density $\rho_0$ is invariant for all nuclei and equals $2.3 \times 10^{17} \text{ kg/m}^3$ . Let $a_{\text{Al}}$ and $a_{\text{Te}}$ be their respective initial accelerations, and $F_{\text{Al}}$ and $F_{\text{Te}}$ be the magnitudes of the buoyant forces exerted on them by the pion fluid. Which of the following represents the correct ratios $a_{\text{Al}} : a_{\text{Te}}$ and $F_{\text{Al}} : F_{\text{Te}}$ ?

Options

A.
$a_{\text{Al}} : a_{\text{Te}} = 1 : 1$ and $F_{\text{Al}} : F_{\text{Te}} = 27 : 125$
B.
$a_{\text{Al}} : a_{\text{Te}} = 125 : 27$ and $F_{\text{Al}} : F_{\text{Te}} = 1 : 1$
C.
$a_{\text{Al}} : a_{\text{Te}} = 1 : 1$ and $F_{\text{Al}} : F_{\text{Te}} = 3 : 5$
D.
$a_{\text{Al}} : a_{\text{Te}} = 27 : 125$ and $F_{\text{Al}} : F_{\text{Te}} = 27 : 125$

Check Your Answer

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

Nuclear Density Invariance|Archimedes' Buoyancy Dynamics|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the Liquid Drop Model combined with Archimedes' principle.

The primary trap is the misapplication of scaling laws: students often confuse the radius scaling ( $R \propto A^{1/3}$ ) with volume scaling, incorrectly concluding that buoyant force scales as $A^{1/3}$ (leading to the $3:5$ trap).

Furthermore, students may incorrectly assume that a heavier nucleus will accelerate differently under a uniform field in a fluid, failing to recognize that both inertial mass and net force scale linearly with volume (and thus $A$ ).

By realizing that nuclear density $\rho_0$ is an intensive property invariant of $A$ , the acceleration becomes purely a function of the density ratio, neutralizing $A$ as a variable.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Density Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free