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title: "A flat circular disk of radius \\(R\\) and total mass \\(M\\) has a non-uniform surface mass density given by \\(\\sigma(r) = \\sigma_0 \\left(1 + \\beta \\dfrac{r^2}{R^2}\\right)\\), where \\(\\sigma_0\\) and \\(\\beta\\) are positive constants and \\(r\\) is the radial distance from the center. The rotational inertia of the disk about an axis perpendicular to the disk through its center is \\[ I = \\left(\\dfrac{3 + 2\\beta}{6 + 3\\beta}\\right) M R^2 \\] Which of the following statements correctly describes and physically justifies the behavior of \\(I\\) in a limiting case?"
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url: "https://nerd-notes.com/ubq/120810/"
date_modified: "2026-08-23T04:43:25+00:00"
---

# A flat circular disk of radius \(R\) and total mass \(M\) has a non-uniform surface mass density given by \(\sigma(r) = \sigma_0 \left(1 + \beta \dfrac{r^2}{R^2}\right)\), where \(\sigma_0\) and \(\beta\) are positive constants and \(r\) is the radial distance from the center. The rotational inertia of the disk about an axis perpendicular to the disk through its center is
\[ I = \left(\dfrac{3 + 2\beta}{6 + 3\beta}\right) M R^2 \]
Which of the following statements correctly describes and physically justifies the behavior of \(I\) in a limiting case?

A flat circular disk of radius \(R\) and total mass \(M\) has a non-uniform surface mass density given by \(\sigma(r) = \sigma_0 \left(1 + \beta \dfrac{r^2}{R^2}\right)\), where \(\sigma_0\) and \(\beta\) are positive constants and \(r\) is the radial distance from the center. The rotational inertia of the disk about an axis perpendicular to the disk through its center is
\[ I = \left(\dfrac{3 + 2\beta}{6 + 3\beta}\right) M R^2 \]
Which of the following statements correctly describes and physically justifies the behavior of \(I\) in a limiting case?

![A circular disk of radius \(R\) lying in a horizontal plane, shown in perspective view as an ellipse. A vertical dashed line passes perpendicularly through the center of the disk, representing the axis of rotation. A thin concentric circular ring of radius \(r\) and differential width \(dr\) is shaded lightly on the surface of the disk. A horizontal solid arrow extends from the central axis to the inner edge of the shaded ring, labeled \(r\). A second solid arrow extends from the central axis to the outer perimeter of the disk, labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460205-TkxyTy.jpg)

- **A.** As \(\beta \to \infty\), \(I \to M R^2\) because the mass is concentrated entirely at the outer rim \(r = R\), behaving like a thin cylindrical hoop.
- **B.** As \(\beta \to 0\), \(I \to \dfrac{1}{2} M R^2\) because the surface mass density becomes uniform across the disk, recovering the rotational inertia of a standard solid disk.
- **C.** As \(\beta \to 0\), \(I \to \dfrac{1}{3} M R^2\) because eliminating the quadratic term concentrates the mass distribution closer to the center than in a uniform disk.
- **D.** As \(\beta \to \infty\), \(I \to \dfrac{1}{2} M R^2\) because the total mass \(M\) is fixed, so changing the density parameter cannot alter the rotational inertia coefficient.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120810/*