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Question #43832

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The Question

A heavy nucleus of rest mass $M$ is firmly held at a distance $R$ from the rotation axis of a large centrifuge. The centrifuge rotates with a constant angular velocity $\omega$ . The nucleus undergoes spontaneous fission into two identical fragments, each of rest mass $m$ . The fragments are constrained to move along a straight, frictionless radial tube fixed to the centrifuge. The rest mass energy released in the fission is $Q = (M - 2m)c^2$ , where $c$ is the speed of light. Assume $\omega R \ll c$ such that classical mechanics applies to the macroscopic motion, but account for the relativistic mass-energy equivalence. Assuming the fission occurs instantaneously, what is the total kinetic energy of the two fragments measured in the rotating frame of the centrifuge immediately after the fission?

Options

A.
$Q$
B.
$Q \left( 1 + \frac{\omega^2 R^2}{2c^2} \right)$
C.
$Q \left( 1 - \frac{\omega^2 R^2}{2c^2} \right)$
D.
$Q \left( 1 - \frac{\omega^2 R^2}{c^2} \right)$

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

Nuclear Fission Energy Balance|Frame of Reference Inertial vs Non-Inertial|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the sophisticated application of the Work-Energy Theorem in non-inertial frames versus inertial frames.

A common trap is to apply standard energy conservation in the inertial frame ( $E_i = E_f$ ) without realizing that the radial tube must exert an impulsive normal force to absorb the angular momentum of the 'missing mass' ( $\Delta m = Q/c^2$ ).

Because the tube is moving in the inertial frame, this impulsive constraint force does negative work, leading to Option B if ignored.

By shifting to the rotating frame, the tube is stationary, its normal force does no work, and the effective Hamiltonian is strictly conserved.

The centrifugal potential energy of the converted mass defect must be subtracted from the Q-value, forcing a frame-dependent effective kinetic energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Fission Energy Balance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free