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Question #43826

Difficulty: 8/10 | 🟢 Free

The Question

In a nuclear physics laboratory, the radius of a $^{27}\text{Al}$ nucleus is measured to be $R = 3.60 \text{ fm}$ with a relative measurement error of $2\%$ . A student is asked to determine the percentage relative error in two specific quantities for this nucleus:

\textbf{Quantity X:} The empirical nuclear density $\rho$ , calculated directly from the measured radius $R$ and the known nuclear mass $M$ (assume the uncertainty in $M$ is negligible).

\textbf{Quantity Y:} The theoretical nuclear volume $V_{th}$ , as dictated by the liquid drop model which assumes perfect saturation of nuclear forces. In this model, the theoretical volume is strictly proportional to the exact integer mass number $A$ (assume the proportionality constant is exactly known).

What are the correct percentage relative errors for Quantity X and Quantity Y, respectively?

Options

A.
$6\%$ and $0\%$
B.
$0\%$ and $6\%$
C.
$6\%$ and $6\%$
D.
$2\%$ and $6\%$

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 Force Range and Saturation|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between empirical error propagation and theoretical constraints imposed by physical models.

The primary trap lies in the cognitive dissonance between the formulas for volume and density.

Students often blindly associate 'volume' with the geometric formula $V = \frac{4}{3}\pi R^3$ , leading them to incorrectly assign a $6\%$ relative error to Quantity Y.

Conversely, they associate 'nuclear density' with the saturation property (constant density), incorrectly assigning a $0\%$ error to Quantity X.

However, Quantity X is explicitly defined as an empirically calculated value derived from the fluctuating measured radius, meaning it must carry the propagated $6\%$ error.

Quantity Y is defined by the theoretical liquid drop model where volume is strictly proportional to the exact integer mass number $A$ ; thus, it is immune to the radial measurement error and has $0\%$ uncertainty.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Force Range and Saturation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free