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Question #43833

Difficulty: 8/10 | 🟢 Free

The Question

A nuclear fission reactor supplies power to an industrial load. The electrical power demanded by the load varies with time as $P(t) = P_0 (1 + \sin(\omega t))^2$ , where $P_0$ is a constant in Watts, $\omega$ is the angular frequency in rad/s, and $t$ is time in seconds. The efficiency of heat transfer from the reactor's thermal output to the electrical load is $\eta$ (where $0 < \eta < 1$ ). Assuming the reactor continuously adjusts its nuclear reaction rate to exactly match the thermal power required, what is the total mass $\Delta m$ consumed by the reactor via mass-energy equivalence ( $E=mc^2$ ) over one complete cycle of the power demand? Let $c$ be the speed of light in vacuum.

Options

A.
$\frac{3 \pi P_0}{\eta \omega c^2}$
B.
$\frac{2 \pi P_0}{\eta \omega c^2}$
C.
$\frac{4 \pi P_0}{\eta \omega c^2}$
D.
$\frac{3 \pi P_0 \eta}{\omega c^2}$

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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

Nuclear Fission Energy Balance|Average Power over Cycle|Work, Energy, and Power

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly integrate a non-linear, time-varying power function and link it to mass-energy equivalence.

The primary trap lies in the treatment of the squared sinusoidal power demand.

Students often incorrectly assume that the average of a squared function is the square of its average, leading them to evaluate the average of $(1 + \sin(\omega t))^2$ as $(1 + 0)^2 = 1$ , which yields the incorrect mass consumption of $2\pi P_0 / (\eta \omega c^2)$ .

Furthermore, it tests the proper application of the efficiency factor $\eta$ ; students must recognize that the reactor must produce more thermal energy than the electrical energy demanded, requiring division by $\eta$ rather than multiplication.

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