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title: "Four identical rigid rotors, initially at rest, each rotate from \\(\\theta = 0\\) to \\(\\theta = \\Theta\\) under the influence of a net torque \\(\\tau(\\theta)\\), where \\(\\tau_0\\) is a positive constant. The torque functions for the four rotors are given in the table.  | Rotor | Net Torque \\(\\tau(\\theta)\\) | |—|—| | 1 | \\(\\tau_0 \\sqrt{\\dfrac{\\theta}{\\Theta}}\\) | | 2 | \\(\\tau_0 \\left(\\dfrac{\\theta}{\\Theta}\\right)  | Rotor 1 | \\(\\tau_0 \\sqrt{\\dfrac{\\theta}{\\Theta}}\\) | | 2 | \\(\\tau_0 \\left(\\dfrac{\\theta}{\\Theta}\\right) | Rotor | Net Torque \\(\\tau(\\theta)\\) | |—|—| | 1 | \\(\\tau_0 \\sqrt{\\dfrac{\\theta}{\\Theta}}\\) | | 2 | \\(\\tau_0 \\left(\\dfrac{\\theta}{\\Theta}\\right)\\) | | 3 | \\(\\tau_0 \\left(\\dfrac{\\theta}{\\Theta}\\right)^2\\) | | 4 | \\(\\tau_0 \\left[1 – \\left(\\dfrac{\\theta}{\\Theta}\\right)^2\\right]\\) |  Which of the following correctly ranks the rotational kinetic energies \\(K_1\\), \\(K_2\\), \\(K_3\\), and \\(K_4\\) of the rotors when they reach \\(\\theta = \\Theta\\)?"
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url: "https://nerd-notes.com/ubq/124237/"
date_modified: "2026-09-28T14:04:24+00:00"
---

# Four identical rigid rotors, initially at rest, each rotate from \(\theta = 0\) to \(\theta = \Theta\) under the influence of a net torque \(\tau(\theta)\), where \(\tau_0\) is a positive constant. The torque functions for the four rotors are given in the table.

| Rotor | Net Torque \(\tau(\theta)\) |
|—|—|
| 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)
 | Rotor 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)
| Rotor | Net Torque \(\tau(\theta)\) |
|—|—|
| 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)\) |
| 3 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)^2\) |
| 4 | \(\tau_0 \left[1 – \left(\dfrac{\theta}{\Theta}\right)^2\right]\) |

Which of the following correctly ranks the rotational kinetic energies \(K_1\), \(K_2\), \(K_3\), and \(K_4\) of the rotors when they reach \(\theta = \Theta\)?

Four identical rigid rotors, initially at rest, each rotate from \(\theta = 0\) to \(\theta = \Theta\) under the influence of a net torque \(\tau(\theta)\), where \(\tau_0\) is a positive constant. The torque functions for the four rotors are given in the table.

| Rotor | Net Torque \(\tau(\theta)\) |
|---|---|
| 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)
 | Rotor 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)
| Rotor | Net Torque \(\tau(\theta)\) |
|---|---|
| 1 | \(\tau_0 \sqrt{\dfrac{\theta}{\Theta}}\) |
| 2 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)\) |
| 3 | \(\tau_0 \left(\dfrac{\theta}{\Theta}\right)^2\) |
| 4 | \(\tau_0 \left[1 - \left(\dfrac{\theta}{\Theta}\right)^2\right]\) |

Which of the following correctly ranks the rotational kinetic energies \(K_1\), \(K_2\), \(K_3\), and \(K_4\) of the rotors when they reach \(\theta = \Theta\)?

- **A.** \(K_3 > K_2 > K_1 > K_4\)
- **B.** \(K_4 > K_1 > K_2 > K_3\)
- **C.** \(K_1 > K_2 > K_3 > K_4\)
- **D.** \(K_1 = K_4 > K_2 > K_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124237/*