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Question #41998

Difficulty: 8/10 | 🟢 Free

The Question

In a theoretical model of a quantum fluid, the fundamental units are Planck's constant $h$ , the speed of light $c$ , and the Planck length $\ell_p$ . The fluid's effective viscosity is postulated to be $\eta = \frac{h}{\ell_p^3}$ . A microscopic sphere of mass $m$ and radius $r$ falls under gravity $g$ in this fluid (ignoring buoyant forces). Due to a velocity-dependent boundary layer at the Planck scale, the drag force deviates from Stokes' law and is given by $F_d = 6\pi \eta r v \left( 1 + \beta \frac{\ell_p v}{r c} \right)$ where $\beta$ is a dimensionless constant and $v$ is the velocity of the sphere. Assuming the perturbation is small ( $\beta \frac{\ell_p v}{r c} \ll 1$ ), what is the first-order correction to the terminal velocity, $\Delta v_t = v_t - v_{t0}$ , where $v_{t0}$ is the unperturbed terminal velocity?

Options

A.
$-\frac{\beta m^2 g^2 \ell_p^7}{36 \pi^2 h^2 r^3 c}$
B.
$-\frac{\beta m^2 g^2 \ell_p^5}{36 \pi^2 h^2 r c}$
C.
$-\frac{\beta m^2 g^2 \ell_p^6}{36 \pi^2 h^2 r^2 c}$
D.
$-\frac{\beta m^2 g^2 \ell_p^8}{36 \pi^2 h^2 r^4 c}$

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

Derived Units via Fundamental Constants|Viscosity and Stokes' Law|Power Series Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-linear perturbations in fluid dynamics while managing derived units of fundamental constants.

A common trap in JEE Advanced is attempting to use dimensional analysis to bypass the calculus or algebra.

This strategy deliberately fails here because $\ell_p$ and $r$ share the same dimension ( $L$ ), making all four options dimensionally identical and valid (all yield dimensions of velocity $L T^{-1}$ ).

Furthermore, students might incorrectly substitute the perturbed velocity linearly or fail to square $v_{t0}$ in the correction term, leading to incorrect powers of $\ell_p$ and $r$ in the denominator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Derived Units via Fundamental Constants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free