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Question #42409

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical space station of radius $R$ rotates with constant angular velocity $\omega$ about its longitudinal axis in deep space. Two identical particles, A and B, each of mass $m$ , are initially at rest on the inner surface. At $t=0$ , particle A is projected radially inwards with a speed $u$ relative to the station, while particle B remains attached to the surface. Both particles act as sources of matter waves. The phase accumulated by a particle is given by $\phi = S/\hbar$ , where $S = \int L dt$ is the classical action evaluated in the inertial frame. What is the magnitude of the phase difference $|\Delta \phi| = |S_A - S_B|/\hbar$ between the two particles at the instant particle A strikes the inner surface again?

Options

A.
$\frac{m R u^3}{\hbar (u^2 + \omega^2 R^2)}$
B.
$\frac{m u^3}{\hbar \omega^2 R}$
C.
$\frac{m u^3}{3 \hbar \omega^2 R}$
D.
$\frac{m R u}{\hbar}$

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

Time of Flight Logic|Phase and Path Difference|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition between non-inertial and inertial frames when evaluating path-dependent integrals like the action.

A common trap is assuming the effective gravity $g_{eff} = \omega^2 R$ is constant and applying standard projectile kinematics ( $T = 2u/g_{eff}$ ) in the rotating frame.

However, the Coriolis and centrifugal forces make the effective 'g' position- and velocity-dependent, rendering standard kinematics invalid.

By shifting to the inertial frame, the trajectory becomes a simple straight line, allowing exact calculation of the time of flight and the action integral, revealing the true phase evolution of the wave packets.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Time of Flight Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free