To solve this problem, we'll break it into parts and use the given values systematically:

Known Values:

• Length of propeller (diameter): $2.10 \, \text{m}$
• Mass of propeller: $110 \, \text{kg}$
• Torque applied: $2250 \, \text{Nm}$
• Revolutions: $5 \, \text{rev}$

The propeller is modeled as a narrow rod rotating about its center. Its moment of inertia is given by:

$I = \frac{1}{12} M L^2$

where $M$ is the mass, and $L$ is the length.

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(a) Angular Acceleration ($\alpha$):

Using the relation: $\alpha = \frac{\tau}{I}$

• Compute $I$: $I = \frac{1}{12} (110) (2.10)^2 = \frac{1}{12} (110) (4.41) = 40.425 \, \text{kg·m}^2$

• Compute $\alpha$: $\alpha = \frac{2250}{40.425} \approx 55.67 \, \text{rad/s}^2$

Answer: $\alpha = 55.67 \, \text{rad/s}^2$

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(b) Angular Velocity After 5 Revolutions ($\omega$):

First, convert $5 \, \text{revolutions}$ into radians: $\theta = 5 \cdot 2\pi = 10\pi \, \text{rad}$

Using the rotational kinematic equation: $\omega^2 = \omega_0^2 + 2\alpha\theta$ where $\omega_0 = 0$ (starting from rest): $\omega^2 = 0 + 2(55.67)(10\pi)$ $\omega^2 = 3498.6\pi \approx 10,992.8$

• Solve for $\omega$: $\omega = \sqrt{10,992.8} \approx 104.86 \, \text{rad/s}$

Answer: $\omega \approx 104.86 \, \text{rad/s}$

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(c) Work Done by the Engine:

The work done by the torque is: $W = \tau \cdot \theta$

Substitute values: $W = 2250 \cdot 10\pi$ $W = 22,500\pi \approx 70,685 \, \text{J}$

Answer: $W \approx 70,685 \, \text{J}$

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(d) Average Power Output of the Engine:

The average power is the work done divided by the time taken. First, find the time using: $\omega = \alpha t$ $t = \frac{\omega}{\alpha} = \frac{104.86}{55.67} \approx 1.88 \, \text{s}$

Now, compute the average power: $P_{\text{avg}} = \frac{W}{t}$ $P_{\text{avg}} = \frac{70,685}{1.88} \approx 37,594 \, \text{W}$

Answer: $P_{\text{avg}} \approx 37,594 \, \text{W}$

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Summary of Results:

1. Angular acceleration: $\alpha = 55.67 \, \text{rad/s}^2$
2. Angular velocity after 5 revolutions: $\omega \approx 104.86 \, \text{rad/s}$
3. Work done: $W \approx 70,685 \, \text{J}$
4. Average power output: $P_{\text{avg}} \approx 37,594 \, \text{W}$

Let me know if you'd like further explanations or clarifications!

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Expansion Questions:

1. How would the results change if the torque was halved?
2. What if the propeller were modeled as a solid disk instead of a narrow rod?
3. How does angular acceleration relate to linear acceleration at the tip of the propeller?
4. What would the instantaneous power output be at the end of the 5 revolutions?
5. How can the moment of inertia be experimentally measured for the propeller?

Tip: Always check if a problem involves energy conservation to simplify torque and work calculations