---
title: "A medical laboratory centrifuge is modeled as a uniform solid disk of mass \\(M\\) and radius \\(R\\) (with rotational inertia \\(I = \\dfrac{1}{2}MR^2\\)). The centrifuge initially rotates counterclockwise at a constant angular speed of \\(N\\) revolutions per minute. At time \\(t = 0\\), the motor is turned off. A constant frictional torque slows the centrifuge down, and it comes to rest after completing exactly \\(n\\) revolutions."
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url: "https://nerd-notes.com/ubq/111400/"
date_modified: "2026-04-11T01:04:13+00:00"
---

# A medical laboratory centrifuge is modeled as a uniform solid disk of mass \(M\) and radius \(R\) (with rotational inertia \(I = \dfrac{1}{2}MR^2\)). The centrifuge initially rotates counterclockwise at a constant angular speed of \(N\) revolutions per minute. At time \(t = 0\), the motor is turned off. A constant frictional torque slows the centrifuge down, and it comes to rest after completing exactly \(n\) revolutions.

A medical laboratory centrifuge is modeled as a uniform solid disk of mass \(M\) and radius \(R\) (with rotational inertia \(I = \dfrac{1}{2}MR^2\)). The centrifuge initially rotates counterclockwise at a constant angular speed of \(N\) revolutions per minute. At time \(t = 0\), the motor is turned off. A constant frictional torque slows the centrifuge down, and it comes to rest after completing exactly \(n\) revolutions.

![A top-down view of a circular disk representing the centrifuge. The disk is shaded gray. A central dot marks the axis of rotation. The radius is labeled with a straight line from the center to the edge labeled 'R'. The mass 'M' is printed on the disk. A curved arrow outside the disk points counterclockwise, labeled 'N' to indicate the initial rotational direction.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775869452-rizUhf.jpg)

**Part a)** **Derive** an expression for the magnitude of the angular acceleration \(\alpha\) of the centrifuge as it slows down. Express your answer in units of \(\text{rad/s}^2\) in terms of \(N\), \(n\), and fundamental constants. *(3 points)*

**Part b)** **Derive** an expression for the magnitude of the frictional torque \(\tau_f\) exerted on the centrifuge as it slows down. Express your answer in terms of \(M\), \(R\), \(N\), \(n\), and fundamental constants. *(2 points)*

**Part c)** A new centrifuge is designed with a rotor that is modeled as a thin hollow hoop of the same mass \(M\) and radius \(R\) (with rotational inertia \(I = MR^2\)). The new centrifuge initially rotates at the same rate of \(N\) revolutions per minute and is subjected to the same constant frictional torque \(\tau_f\) when the motor is turned off. **Indicate** whether the number of revolutions it takes for the new centrifuge to come to rest is greater than, less than, or equal to \(n\). - [ ] Greater than \(n\) - [ ] Less than \(n\) - [ ] Equal to \(n\) **Justify** your answer using physical principles. *(3 points)*

**Part d)** **Sketch** a graph of the angular velocity \(\omega\) as a function of time \(t\) for the original solid disk centrifuge from \(t = 0\) until it comes to rest. On the same set of axes, **sketch** a graph of \(\omega\) as a function of time \(t\) for the new hollow hoop centrifuge from \(t = 0\) until it comes to rest. Clearly label the two lines 'Disk' and 'Hoop'. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/111400/*