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

 

[kyle8921]

Based on the given information, the average power output of the engine during the first 5.00 revolutions of the propeller can be calculated using the formula P = τω, where τ is the applied torque and ω is the angular velocity. We can first calculate the angular acceleration (α) using the formula α = τ/I, where I is the moment of inertia of the propeller. In this case, I = 42.18 kg*m^2. After solving for α, we can use the formula ω = ω0 + αt to find the angular velocity after 5 revolutions. ω0 is the initial angular velocity, which is 0 since the propeller starts from rest. We can then substitute the values into P = τω and solve for the average power output.

For the second question, the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 revolutions can be calculated using the formula P = dW/dt, where W is the work done by the engine on the propeller. We can first calculate the work done after 5 revolutions using the formula W = τθ, where θ is the angle through which the propeller has turned (5 revolutions = 10π radians). Then, we can take the derivative of W with respect to time (t) to find the instantaneous power output. Keep in mind that the units for power are watts (W), which can be converted from Nm/s or J/s.

It seems like you were on the right track with your methods, but the units may have been incorrect. It's always a good idea to double check your units and make sure they are consistent throughout your calculations. I hope this helps and good luck with your calculations!