---
title: "A uniform thin rod of rotational inertia \\(I\\) is pivoted at one end and rotates in a horizontal plane about a vertical axis through the pivot. The rod is immersed in a viscous fluid that exerts a retarding torque of magnitude \\(\\tau = b\\omega\\) on the rod, where \\(b\\) is a positive constant and \\(\\omega\\) is the instantaneous angular speed. At time \\(t = 0\\), the rod has an initial angular speed \\(\\omega_0\\) and rotational kinetic energy \\(K_0\\). Which of the following expressions correctly represents the kinetic energy \\(K\\) of the rod as a function of its angular displacement \\(\\theta\\) from its initial position before it comes to rest?"
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url: "https://nerd-notes.com/ubq/124336/"
date_modified: "2026-09-28T14:04:54+00:00"
---

# A uniform thin rod of rotational inertia \(I\) is pivoted at one end and rotates in a horizontal plane about a vertical axis through the pivot. The rod is immersed in a viscous fluid that exerts a retarding torque of magnitude \(\tau = b\omega\) on the rod, where \(b\) is a positive constant and \(\omega\) is the instantaneous angular speed. At time \(t = 0\), the rod has an initial angular speed \(\omega_0\) and rotational kinetic energy \(K_0\). Which of the following expressions correctly represents the kinetic energy \(K\) of the rod as a function of its angular displacement \(\theta\) from its initial position before it comes to rest?

A uniform thin rod of rotational inertia \(I\) is pivoted at one end and rotates in a horizontal plane about a vertical axis through the pivot. The rod is immersed in a viscous fluid that exerts a retarding torque of magnitude \(\tau = b\omega\) on the rod, where \(b\) is a positive constant and \(\omega\) is the instantaneous angular speed. At time \(t = 0\), the rod has an initial angular speed \(\omega_0\) and rotational kinetic energy \(K_0\). Which of the following expressions correctly represents the kinetic energy \(K\) of the rod as a function of its angular displacement \(\theta\) from its initial position before it comes to rest?

![A top-down view of a horizontal plane showing a thin straight rod pivoted at its left end. A fixed pivot is represented by a small solid black circle at the left end of the rod. The uniform rod extends horizontally to the right from the pivot to a free end. Above the free end, a curved arrow pointing counterclockwise indicates the direction of motion and is labeled \(\omega\). Below the rod, a curved arrow pointing clockwise indicates the opposing retarding torque and is labeled \(\tau\). A light gray bounding box encloses the rod to represent the fluid bath. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604294-GQXzMN.jpg)

- **A.** \(K(\theta) = K_0 e^{-2b\theta / (I\omega_0)}\)
- **B.** \(K(\theta) = K_0 \left(1 - \dfrac{b}{I\omega_0}\theta\right)\)
- **C.** \(K(\theta) = K_0 \left(1 - \dfrac{b}{I\omega_0}\theta\right)^2\)
- **D.** \(K(\theta) = K_0 \left(1 - \dfrac{2b}{I\omega_0}\theta\right)^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124336/*