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title: "A uniform horizontal disk of mass \\(M\\) and radius \\(R\\) rotates freely about a vertical, frictionless axle through its center at an initial angular velocity \\(\\omega_0\\). A thin, non-uniform rod of mass \\(M\\) and length \\(R\\) has a linear mass density given by \\(\\lambda(r) = C r\\), where \\(r\\) is the distance from one end of the rod and \\(C\\) is a constant. The rod, initially at rest, is dropped vertically onto the spinning disk such that the end with \\(r = 0\\) lands precisely at the disk’s center and the rod lies radially along the disk surface. Friction between the disk and the rod quickly brings the rod to rest relative to the disk. What is the final angular velocity of the disk-rod system?"
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url: "https://nerd-notes.com/ubq/117675/"
date_modified: "2026-08-04T07:49:53+00:00"
---

# A uniform horizontal disk of mass \(M\) and radius \(R\) rotates freely about a vertical, frictionless axle through its center at an initial angular velocity \(\omega_0\). A thin, non-uniform rod of mass \(M\) and length \(R\) has a linear mass density given by \(\lambda(r) = C r\), where \(r\) is the distance from one end of the rod and \(C\) is a constant. The rod, initially at rest, is dropped vertically onto the spinning disk such that the end with \(r = 0\) lands precisely at the disk’s center and the rod lies radially along the disk surface. Friction between the disk and the rod quickly brings the rod to rest relative to the disk. What is the final angular velocity of the disk-rod system?

A uniform horizontal disk of mass \(M\) and radius \(R\) rotates freely about a vertical, frictionless axle through its center at an initial angular velocity \(\omega_0\). A thin, non-uniform rod of mass \(M\) and length \(R\) has a linear mass density given by \(\lambda(r) = C r\), where \(r\) is the distance from one end of the rod and \(C\) is a constant. The rod, initially at rest, is dropped vertically onto the spinning disk such that the end with \(r = 0\) lands precisely at the disk's center and the rod lies radially along the disk surface. Friction between the disk and the rod quickly brings the rod to rest relative to the disk. What is the final angular velocity of the disk-rod system?

![A 3D perspective diagram showing a horizontal circular disk of radius R centered on a vertical dashed axis line z. The disk is rotating counterclockwise as viewed from above, indicated by a curved arrow labeled \omega_0 around the central axis. Above the disk, a thin rod of length R is oriented horizontally along a radial line extending from the center axis out to radius R. A downward arrow indicates the rod falling onto the disk. The rod is shaded with a gradient that becomes darker toward its outer end to represent increasing density \lambda(r) = Cr. Labels show mass M and radius R for the disk, and mass M and length R for the rod. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829793-YhBl8I.jpg)

- **A.** \(\dfrac{1}{3} \omega_0\)
- **B.** \(\dfrac{1}{2} \omega_0\)
- **C.** \(\dfrac{3}{5} \omega_0\)
- **D.** \(\dfrac{1}{\sqrt{2}} \omega_0\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117675/*