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Question #43766

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid cylinder of initial mass $M_0$ and initial radius $R_0$ is initially at rest in free space, free to rotate about its fixed central longitudinal axis. A laser beam of constant power $P$ is continuously adjusted so that it always strikes the curved surface of the cylinder tangentially. The cylinder perfectly absorbs the laser photons, and the resulting radiation pressure exerts a torque.

The absorbed laser energy causes the material at the edge to ablate (vaporize) continuously. The latent heat of ablation of the material is $L$ (energy required to ablate unit mass). The ablated gas is ejected radially outwards in the rotating frame of the cylinder. As mass is lost, the cylinder remains a uniform solid cylinder of constant length, but its radius $R(t)$ decreases.

What is the angular velocity $\omega$ of the cylinder at the instant its mass reduces to $\frac{M_0}{2}$ ?

(Given: $c$ is the speed of light. Ignore any relativistic mass corrections for the cylinder.)

Options

A.
$\frac{2L}{c R_0}$
B.
$\frac{4L}{3c R_0}(4 - \sqrt{2})$
C.
$\frac{4L}{c R_0}(\sqrt{2} - 1)$
D.
$\frac{2L \ln 2}{c R_0}$

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

Intensity and Photon Flux|Variable Mass Systems|Angular Momentum conservation

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the angular impulse-momentum theorem in variable mass systems.

A common trap is blindly applying $\tau_{ext} = \frac{d}{dt}(I\omega)$ to the cylinder alone, forgetting that the system boundary must include the departing ablated mass which carries away angular momentum.

Students who make this error will incorrectly include an extra $-\dot{I}\omega$ term, leading to a flawed differential equation (Option B).

Another trap is failing to recognize that since ablation occurs at the curved surface, the radius $R(t)$ must decrease to conserve the cylinder's uniform density and constant length, leading to incorrect constant-radius assumptions (Options A and D).

The problem elegantly demonstrates that when specific angular momentum of departing mass equals $\dot{I}/\dot{m}$ , the effective torque equation simplifies to $\tau = I\dot{\omega}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Intensity and Photon Flux
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free