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Question #42355

Difficulty: 8/10 | 🟢 Free

The Question

A microscopic cavity contains $N$ molecules of an ideal rigid diatomic gas, each of mass $m$ . The cavity is driven by an external mechanism to oscillate along the $x$ -axis with a velocity $v(t) = V_0 \cos(\omega t)$ . The frequency $\omega$ is sufficiently high such that the center of mass of the gas oscillates synchronously with the cavity. In a dynamic steady state, the time-averaged total kinetic energy of the gas (including both translational and rotational motion) measured in the stationary laboratory frame is $E$ . Which of the following expressions gives the true thermodynamic temperature $T$ of the gas? (Here $k_B$ is the Boltzmann constant).

Options

A.
$\frac{4E - N m V_0^2}{10 N k_B}$
B.
$\frac{2E}{5 N k_B}$
C.
$\frac{4E + N m V_0^2}{10 N k_B}$
D.
$\frac{4E - N m V_0^2}{6 N k_B}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Center of Mass (COM) Frame|Energy of Vibrational Modes|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the macroscopic bulk kinetic energy of a system and its microscopic internal energy (thermal energy) in a non-inertial, oscillating reference frame.

A common trap is to equate the total time-averaged kinetic energy measured in the lab frame directly to the internal energy ( $E = \frac{5}{2} N k_B T$ ), implicitly treating the cavity as a static boundary.

By forcing the transition from the lab frame to the center-of-mass frame, students must correctly apply the kinetic energy decomposition theorem ( $K_{lab} = K_{com} + K_{internal}$ ) and compute the time-averaged bulk kinetic energy ( $\frac{1}{4} N m V_0^2$ ) to isolate the true thermodynamic temperature.

An additional trap is using the incorrect degrees of freedom for a diatomic gas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Center of Mass (COM) Frame
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free