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Question #42585

Difficulty: 8/10 | 🟢 Free

The Question

A closed cylindrical tube of length $L$ and uniform cross-sectional area contains an ideal gas of molar mass $M$ . The tube is accelerated uniformly with an acceleration $a$ along its longitudinal axis. The gas is maintained at a constant thermodynamic temperature $T$ . If $u_f$ and $u_r$ represent the translational kinetic energy per unit volume of the gas at the front (leading) and rear (trailing) ends of the tube respectively, which of the following expressions gives the ratio $\frac{u_r}{u_f}$ ? (Assume $R$ is the universal gas constant).

Options

A.
$\exp\left(\frac{M a L}{R T}\right)$
B.
$\exp\left(\frac{3 M a L}{2 R T}\right)$
C.
$\exp\left(\frac{2 M a L}{3 R T}\right)$
D.
$1 + \frac{3 M a L}{2 R T}$

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

Pressure-Kinetic Energy Relation|Accelerated Frame Hydrostatics|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple macroscopic non-inertial hydrostatics with microscopic kinetic theory.

A common trap is assuming the gas density remains constant, leading to a linear pressure gradient and the incorrect option D.

Another sophisticated trap exploits the relation $P = \frac{2}{3}u$ ; students often erroneously substitute this into the density equation without properly canceling the coefficients in the differential equation, leading them to incorrectly inject a $\frac{3}{2}$ or $\frac{2}{3}$ factor into the exponential argument (Options B and C).

The correct approach recognizes that the proportionality constant cancels out when taking the ratio.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Pressure-Kinetic Energy Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free