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title: "A uniform wheel of rotational inertia \\(I\\) is mounted on a fixed horizontal axle through its center. Starting from rest at angular position \\(\\theta = 0\\), a mechanism exerts a net torque such that the wheel’s angular acceleration varies with angular displacement according to \\(\\alpha(\\theta) = b\\theta^2\\), where \\(b\\) is a positive constant. A student uses the constant-acceleration kinematic equation \\(\\omega^2 = \\omega_0^2 + 2\\alpha\\Delta\\theta\\) to estimate the angular speed at displacement \\(\\theta\\), substituting \\(\\alpha(\\theta) = b\\theta^2\\) to obtain \\(\\omega_{\\text{est}} = \\sqrt{2b\\theta^3}\\). Which of the following statements correctly compares the actual angular speed \\(\\omega\\) to \\(\\omega_{\\text{est}}\\) and provides the correct justification?"
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url: "https://nerd-notes.com/ubq/124430/"
date_modified: "2026-09-28T14:05:29+00:00"
---

# A uniform wheel of rotational inertia \(I\) is mounted on a fixed horizontal axle through its center. Starting from rest at angular position \(\theta = 0\), a mechanism exerts a net torque such that the wheel’s angular acceleration varies with angular displacement according to \(\alpha(\theta) = b\theta^2\), where \(b\) is a positive constant. A student uses the constant-acceleration kinematic equation \(\omega^2 = \omega_0^2 + 2\alpha\Delta\theta\) to estimate the angular speed at displacement \(\theta\), substituting \(\alpha(\theta) = b\theta^2\) to obtain \(\omega_{\text{est}} = \sqrt{2b\theta^3}\). Which of the following statements correctly compares the actual angular speed \(\omega\) to \(\omega_{\text{est}}\) and provides the correct justification?

A uniform wheel of rotational inertia \(I\) is mounted on a fixed horizontal axle through its center. Starting from rest at angular position \(\theta = 0\), a mechanism exerts a net torque such that the wheel's angular acceleration varies with angular displacement according to \(\alpha(\theta) = b\theta^2\), where \(b\) is a positive constant. A student uses the constant-acceleration kinematic equation \(\omega^2 = \omega_0^2 + 2\alpha\Delta\theta\) to estimate the angular speed at displacement \(\theta\), substituting \(\alpha(\theta) = b\theta^2\) to obtain \(\omega_{\text{est}} = \sqrt{2b\theta^3}\). Which of the following statements correctly compares the actual angular speed \(\omega\) to \(\omega_{\text{est}}\) and provides the correct justification?

![A circular disk of radius R is shown in a front view, centered on a small black circular axle. A horizontal dashed line extends radially to the right from the center of the axle to the outer rim of the disk, representing the initial angular position at zero radians. A solid radial line segment extends from the center of the axle upward and to the right at an acute angle to the outer rim. A curved, counterclockwise arrow spans from the horizontal dashed line to the solid radial line segment, labeled with the symbol \theta. A second curved counterclockwise arrow is positioned just outside the upper-right perimeter of the disk, labeled \alpha(\theta). A small curved arrow near the axle indicates the direction of rotation. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604329-XtybSO.jpg)

- **A.** The actual angular speed is greater than \(\omega_{\text{est}}\) because the angular acceleration increases with angular displacement, so the angular speed increases at an increasing rate rather than at a constant rate.
- **B.** The actual angular speed is equal to \(\omega_{\text{est}}\) because the kinematic equation \(\omega^2 = \omega_0^2 + 2\alpha\Delta\theta\) is valid for any rotational motion as long as the instantaneous angular acceleration evaluated at the final displacement is used.
- **C.** The actual angular speed is less than \(\omega_{\text{est}}\) because \(\alpha\) varies with \(\theta\), and applying the chain rule \(\alpha = \omega\dfrac{d\omega}{d\theta}\) yields \(\int_0^\omega \omega'\,d\omega' = \int_0^\theta b(\theta')^2\,d\theta'\), which results in \(\omega = \sqrt{\dfrac{2}{3}b\theta^3}\).
- **D.** The actual angular speed is less than \(\omega_{\text{est}}\) because the angular acceleration varies from \(0\) to \(b\theta^2\), so substituting the average angular acceleration \(\alpha_{\text{avg}} = \dfrac{1}{2}b\theta^2\) into the kinematic equation yields \(\omega = \sqrt{2\alpha_{\text{avg}}\theta} = \sqrt{b\theta^3}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124430/*