AP Physics C: Mechanics
6.2 Torque and Work
6.1 Rotational Kinetic Energy
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.
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 \(\theta = \Theta\)?
| 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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