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title: "A uniform solid star of mass \\(M\\) and initial radius \\(R_0\\) rotates about an axis through its center with initial angular velocity \\(\\omega_0\\). The star undergoes core collapse to a final radius \\(R_f\\), during which its mass distribution shifts to a non-uniform density profile given by \\(\\rho(r) = \\dfrac{C}{r}\\) for \\(0 \\le r \\le R_f\\), where \\(C\\) is a constant. Assuming no external torques act on the star, what is the final angular velocity \\(\\omega_f\\) of the star in terms of \\(\\omega_0\\), \\(R_0\\), and \\(R_f\\)?"
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url: "https://nerd-notes.com/ubq/117637/"
date_modified: "2026-08-04T07:49:26+00:00"
---

# A uniform solid star of mass \(M\) and initial radius \(R_0\) rotates about an axis through its center with initial angular velocity \(\omega_0\). The star undergoes core collapse to a final radius \(R_f\), during which its mass distribution shifts to a non-uniform density profile given by \(\rho(r) = \dfrac{C}{r}\) for \(0 \le r \le R_f\), where \(C\) is a constant. Assuming no external torques act on the star, what is the final angular velocity \(\omega_f\) of the star in terms of \(\omega_0\), \(R_0\), and \(R_f\)?

A uniform solid star of mass \(M\) and initial radius \(R_0\) rotates about an axis through its center with initial angular velocity \(\omega_0\). The star undergoes core collapse to a final radius \(R_f\), during which its mass distribution shifts to a non-uniform density profile given by \(\rho(r) = \dfrac{C}{r}\) for \(0 \le r \le R_f\), where \(C\) is a constant. Assuming no external torques act on the star, what is the final angular velocity \(\omega_f\) of the star in terms of \(\omega_0\), \(R_0\), and \(R_f\)?

![Two spherical bodies placed side-by-side representing the initial and final states of a star. On the left, a larger sphere with radius \(R_0\) is shaded uniformly in light grey. A curved arrow encircling its vertical axis points counterclockwise, labeled \(\omega_0\). A horizontal dashed line segment extends from the center to the right outer edge, labeled \(R_0\). An arrow pointing right connects the left sphere to the right sphere, representing collapse. On the right, a smaller sphere with radius \(R_f\) has a radial shading gradient that is darkest at the center and fades lighter toward the outer boundary. A curved arrow encircling its vertical axis points counterclockwise, labeled \(\omega_f\). A horizontal dashed line segment extends from its center to its outer edge, labeled \(R_f\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829765-UA4KXD.jpg)

- **A.** \(\left(\dfrac{R_0}{R_f}\right)^2 \omega_0\)
- **B.** \(\dfrac{3}{5} \left(\dfrac{R_0}{R_f}\right)^2 \omega_0\)
- **C.** \(\dfrac{5}{6} \left(\dfrac{R_0}{R_f}\right)^2 \omega_0\)
- **D.** \(\dfrac{6}{5} \left(\dfrac{R_0}{R_f}\right)^2 \omega_0\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117637/*