Before You Begin

Question #43664

Difficulty: 8/10 | 🟢 Free

The Question

A diatomic molecule is modeled as two point masses $m_1 = m \text{ (kg)}$ and $m_2 = 2m \text{ (kg)}$ connected by a massless spring of force constant $k \text{ (N/m)}$ and natural length $r_0 \text{ (m)}$ . The molecule rotates freely in space with a constant angular momentum $J \text{ (kg}\cdot\text{m}^2\text{/s)}$ about an axis passing through its center of mass and perpendicular to the internuclear axis. Due to the rotation, the equilibrium separation between the masses extends to $r_e \text{ (m)}$ . What is the angular frequency $\omega \text{ (rad/s)}$ of small radial oscillations of the masses about this new equilibrium separation $r_e$ ?

Options

A.
$\sqrt{\frac{3k}{2m} + \frac{27J^2}{4m^2 r_e^4}}$
B.
$\sqrt{\frac{3k}{2m}}$
C.
$\sqrt{\frac{3k}{2m} + \frac{9J^2}{4m^2 r_e^4}}$
D.
$\sqrt{\frac{k}{3m} + \frac{3J^2}{m^2 r_e^4}}$

Check Your Answer

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

Reduced Mass Effect|Torsional Oscillations|Center of Mass (COM) Frame|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate rotational dynamics with simple harmonic motion using the effective potential method.

The primary trap is treating the vibrational degrees of freedom as a simple 1D harmonic oscillator ( $\omega = \sqrt{k/\mu}$ ) while ignoring the cross-coupling term that arises from the conservation of angular momentum.

When the molecule stretches, its moment of inertia increases, which must decrease its angular velocity to conserve $J$ .

This rotational kinetic energy constraint acts as an additional 'centrifugal spring', stiffening the effective restoring force.

Students who fail to account for the second derivative of the centrifugal potential term will incorrectly choose the standard SHM frequency or miss the factor of 3 in the coupling term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Reduced Mass Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free