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title: "A rigid rotor with rotational inertia \\(I\\) is initially at rest at angular position \\(\\theta = 0\\). In three separate trials, the rotor is turned from \\(\\theta = 0\\) to \\(\\theta = 1\\text{ rad}\\) by a single applied torque: Trial 1 uses constant torque \\(\\tau_1(\\theta) = \\tau_0\\), Trial 2 uses linearly increasing torque \\(\\tau_2(\\theta) = \\tau_0\\theta\\), and Trial 3 uses sinusoidal torque \\(\\tau_3(\\theta) = \\tau_0\\sin\\left(\\dfrac{\\pi}{2}\\theta\\right)\\). Which row in the table correctly identifies the work delivered in Trial 2 (\\(W_2\\)), the work delivered in Trial 3 (\\(W_3\\)), and the ranking of the final rotational kinetic energies \\(K_1, K_2, K_3\\) of the rotor at \\(\\theta = 1\\text{ rad}\\)?"
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url: "https://nerd-notes.com/ubq/124327/"
date_modified: "2026-09-28T14:04:52+00:00"
---

# A rigid rotor with rotational inertia \(I\) is initially at rest at angular position \(\theta = 0\). In three separate trials, the rotor is turned from \(\theta = 0\) to \(\theta = 1\text{ rad}\) by a single applied torque: Trial 1 uses constant torque \(\tau_1(\theta) = \tau_0\), Trial 2 uses linearly increasing torque \(\tau_2(\theta) = \tau_0\theta\), and Trial 3 uses sinusoidal torque \(\tau_3(\theta) = \tau_0\sin\left(\dfrac{\pi}{2}\theta\right)\). Which row in the table correctly identifies the work delivered in Trial 2 (\(W_2\)), the work delivered in Trial 3 (\(W_3\)), and the ranking of the final rotational kinetic energies \(K_1, K_2, K_3\) of the rotor at \(\theta = 1\text{ rad}\)?

A rigid rotor with rotational inertia \(I\) is initially at rest at angular position \(\theta = 0\). In three separate trials, the rotor is turned from \(\theta = 0\) to \(\theta = 1\text{ rad}\) by a single applied torque: Trial 1 uses constant torque \(\tau_1(\theta) = \tau_0\), Trial 2 uses linearly increasing torque \(\tau_2(\theta) = \tau_0\theta\), and Trial 3 uses sinusoidal torque \(\tau_3(\theta) = \tau_0\sin\left(\dfrac{\pi}{2}\theta\right)\). Which row in the table correctly identifies the work delivered in Trial 2 (\(W_2\)), the work delivered in Trial 3 (\(W_3\)), and the ranking of the final rotational kinetic energies \(K_1, K_2, K_3\) of the rotor at \(\theta = 1\text{ rad}\)?

![A shape-focused graph with bare axes showing net torque \(\tau\) on the vertical axis versus angular displacement \(\theta\) on the horizontal axis. The horizontal axis has a tick mark at \(0\) at the origin and a tick mark at \(1\) at the right. The vertical axis has a tick mark at \(0\) at the origin and a tick mark at \(\tau_0\) near the top. Three distinct curves span from \(\theta = 0\) to \(\theta = 1\). The first curve is a horizontal solid line at height \(\tau_0\) running from \(\theta = 0\) to \(\theta = 1\), labeled 1. The second curve is a straight dashed line from the origin \((0,0)\) to the point \((1, \tau_0)\), labeled 2. The third curve is a dotted line starting at \((0,0)\) that curves upward concavely downward to meet the point \((1, \tau_0)\) with zero slope, labeled 3. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604291-ol0lNS.jpg)

- **A.** | \(W_2 = \dfrac{1}{2}\tau_0\) | \(W_3 = \dfrac{2}{\pi}\tau_0\) | \(K_2 < K_3 < K_1\) |
- **B.** | \(W_2 = \tau_0\) | \(W_3 = \dfrac{\pi}{2}\tau_0\) | \(K_1 = K_2 = K_3\) |
- **C.** | \(W_2 = \dfrac{1}{2}\tau_0\) | \(W_3 = \dfrac{\pi}{2}\tau_0\) | \(K_2 < K_1 < K_3\) |
- **D.** | \(W_2 = \tau_0\) | \(W_3 = \dfrac{2}{\pi}\tau_0\) | \(K_3 < K_2 < K_1\) |

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124327/*