Solve Propeller Problem: Average & Instantaneous Power Output

[Kenchin]

An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airplane's engine is first started, it applies a constant torque of 1590Nm to the propeller, which starts from rest.

Question I:
What is the average power output of the engine during the first 5.00 rev?

Question II:
What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 rev?


I've already solved for the angular accelleration (after 5 revolutions) which is alpha, angular speed omega (after 5 revolutions), and work after 5 revolutions W, moment of inertia 42.18kg*m^2.

For the last two parts I've tried to solve using P=torque+angular velocity ... that turned out to be wrong. Then I tried using P=Change in work/change in time but that failed. So now I'm a little at a loss. Is there any suggestions where to try next?

I figured it out, my methods were correct... my ending units were wrong! @_@

 

[Andrew Mason]

An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airplane's engine is first started, it applies a constant torque of 1590Nm to the propeller, which starts from rest.

Question I:
What is the average power output of the engine during the first 5.00 rev?

Question II:
What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 rev?I've already solved for the angular accelleration (after 5 revolutions) which is alpha, angular speed omega (after 5 revolutions), and work after 5 revolutions W, moment of inertia 42.18kg*m^2.

For the last two parts I've tried to solve using P=torque+angular velocity ... that turned out to be wrong. Then I tried using P=Change in work/change in time but that failed. So now I'm a little at a loss. Is there any suggestions where to try next?

I figured it out, my methods were correct... my ending units were wrong! @_@

Energy is torque x angle (force x distance).

$$\tau\Delta\theta = \text{Work}$$

So $$P_{avg} = \Delta E/\Delta t = \tau\Delta\theta/\Delta t$$

All you have to do is figure out how long it takes to move the propeller 5 revolutions with that torque: Use $\theta = \frac{1}{2}\alpha t^2$ and $\alpha = \tau/I$ to find the time in terms of angle and torque (and I).

To find instantaneous power, use:

$$P = \tau\omega = \tau\alpha\Delta t$$

You have to assume that in the first 5 revolutions, the resistance to motion is only the moment of inertia of the propeller, not the propulsion of air by the propeller.

AM