---
title: "A disk rotates about a fixed axis perpendicular to its center. The disk’s angular velocity \\(\\omega\\) as a function of time \\(t\\) is shown in the graph and modeled by \\(\\omega(t) = ct^2\\), where \\(c\\) is a positive constant. At \\(t = 4.0\\text{ s}\\), the angular velocity is \\(24\\text{ rad/s}\\), and a dashed line tangent to the curve at \\(t = 2.0\\text{ s}\\) has a slope of \\(6.0\\text{ rad/s}^2\\). What is the total angular displacement of the disk between \\(t = 0\\text{ s}\\) and \\(t = 4.0\\text{ s}\\)?"
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url: "https://nerd-notes.com/ubq/124389/"
date_modified: "2026-09-28T14:05:08+00:00"
---

# A disk rotates about a fixed axis perpendicular to its center. The disk’s angular velocity \(\omega\) as a function of time \(t\) is shown in the graph and modeled by \(\omega(t) = ct^2\), where \(c\) is a positive constant. At \(t = 4.0\text{ s}\), the angular velocity is \(24\text{ rad/s}\), and a dashed line tangent to the curve at \(t = 2.0\text{ s}\) has a slope of \(6.0\text{ rad/s}^2\). What is the total angular displacement of the disk between \(t = 0\text{ s}\) and \(t = 4.0\text{ s}\)?

A disk rotates about a fixed axis perpendicular to its center. The disk's angular velocity \(\omega\) as a function of time \(t\) is shown in the graph and modeled by \(\omega(t) = ct^2\), where \(c\) is a positive constant. At \(t = 4.0\text{ s}\), the angular velocity is \(24\text{ rad/s}\), and a dashed line tangent to the curve at \(t = 2.0\text{ s}\) has a slope of \(6.0\text{ rad/s}^2\). What is the total angular displacement of the disk between \(t = 0\text{ s}\) and \(t = 4.0\text{ s}\)?

![A Cartesian coordinate graph showing a horizontal axis labeled t (s) marked with major tick marks at 0, 1, 2, 3, and 4, and a vertical axis labeled \omega (rad/s) marked with major tick marks at 0, 6, 12, 18, and 24. Light gray gridlines form a uniform square grid across the plot area from t = 0 to 4 and \omega = 0 to 24. A thick solid black curve starts at the origin (0, 0), curves upward with increasing slope, passes through the points (2, 6) and (4, 24), and terminates at (4, 24). A straight dashed black line is tangent to the curve at the point (2, 6); this dashed line extends from (1, 0) on the horizontal axis through (2, 6) to (3, 12). A solid black dot marks the point of tangency at (2, 6). No other labels, curves, lines, text, or background shading appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604307-5aTw3g.jpg)

- **A.** \(12\text{ rad}\)
- **B.** \(32\text{ rad}\)
- **C.** \(48\text{ rad}\)
- **D.** \(96\text{ rad}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124389/*