Propeller Rotation Problem - Please check

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Homework Help Overview

The problem involves an airplane propeller with a specified length and mass, subjected to a constant torque as it starts from rest. Participants are tasked with calculating angular acceleration, angular speed after a set number of revolutions, work done, average power, and instantaneous power output.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the moment of inertia and its calculation, with one noting a potential error in the original poster's values. There are attempts to clarify the formulas used for angular acceleration, work done, and average power. Questions arise regarding the correctness of the calculations and the interpretation of formulas.

Discussion Status

Some participants have provided feedback on the calculations, pointing out possible mistakes and clarifying the use of formulas. There is an ongoing exploration of the correct approach to average power, with acknowledgment of the original poster's oversight in squaring the angular speed.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the extent of guidance provided. There is an emphasis on ensuring understanding of the underlying concepts rather than simply providing answers.

JWDavid
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Thanks in advance - it went smoothly so I'm hoping I'm right but... usually that's when I've made a big mistake.

Homework Statement


An airplane propeller is 2.08 m (tip to tip) and has a mass of 117 kg. When the engine is first started it applies a constant torque of 1950 N m to the propeller which starts from rest.
a) What is the angular acceleration of the propeller? (model it on a slender rod).
b) What is the propeller's angular speed after 5.00 revolutions?
c) How much work is done during the first 5.00 revolutions by the engine?
d) What is the average power during the first 5.00 revolutions?
e) What is the instantaneous power output of the engine the instant it has completed 5.00 revolutions?


Homework Equations


I think this is all of them
I = mL2/12
\Thetaf = \Thetai + \omegat + \alphat2/2
\omegaf = \omegai + \alphat
W = \DeltaKE = I\omega2/2
Pavg = \DeltaP/2 = \tau\omega/2
Pins = \tau\omega


The Attempt at a Solution


work that got me to the answers below.
a) 42.2 b) 53.9 rad/sec2 c) 1.14 kJ d) 52.6 kW e) 10.5 kW


I = mL2/12 = 117*2.082/12 = 42.2

a) \tau = 1950 N m = I\alpha
\alpha = 1950/42.2 = 46.2 rad/sec2

b) \Thetaf = \Thetai + \omegat + \alphat2/2
5*2\pi = 0 + 0 + 46.2t2/2

t = squareroot (20\pi/46.2) = 1.166 seconds

\omegaf = \omegai + \alphat
= 0 + 46.2*1.166 = 53.9 rad/sec

c) W = \DeltaKE = I\omega2/2
= 42.2*53.9/2 = 1.14kJ

d) Pavg = \DeltaP/2 = \tau\omega/2 = 1950*53.9/2 = 52.6 kW

e) Pins = \tau\omega = 1950*53.9 = 10.5 kW/s
 
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Hi JWDavid,

JWDavid said:
Thanks in advance - it went smoothly so I'm hoping I'm right but... usually that's when I've made a big mistake.

Homework Statement


An airplane propeller is 2.08 m (tip to tip) and has a mass of 117 kg. When the engine is first started it applies a constant torque of 1950 N m to the propeller which starts from rest.
a) What is the angular acceleration of the propeller? (model it on a slender rod).
b) What is the propeller's angular speed after 5.00 revolutions?
c) How much work is done during the first 5.00 revolutions by the engine?
d) What is the average power during the first 5.00 revolutions?
e) What is the instantaneous power output of the engine the instant it has completed 5.00 revolutions?


Homework Equations


I think this is all of them
I = mL2/12
\Thetaf = \Thetai + \omegat + \alphat2/2
\omegaf = \omegai + \alphat
W = \DeltaKE = I\omega2/2
Pavg = \DeltaP/2 = \tau\omega/2
Pins = \tau\omega


The Attempt at a Solution


work that got me to the answers below.
a) 42.2

I just wanted to point out here that you seemed to have typed in the wrong value; 42.2 is the moment of inertia; but I see you have the correct answer for the angular acceleration below.

b) 53.9 rad/sec2 c) 1.14 kJ d) 52.6 kW e) 10.5 kW


I = mL2/12 = 117*2.082/12 = 42.2

a) \tau = 1950 N m = I\alpha
\alpha = 1950/42.2 = 46.2 rad/sec2

b) \Thetaf = \Thetai + \omegat + \alphat2/2
5*2\pi = 0 + 0 + 46.2t2/2

t = squareroot (20\pi/46.2) = 1.166 seconds

\omegaf = \omegai + \alphat
= 0 + 46.2*1.166 = 53.9 rad/sec

c) W = \DeltaKE = I\omega2/2
= 42.2*53.9/2 = 1.14kJ

I don't think this is correct; it looks like you neglected to square the angular speed.

d) Pavg = \DeltaP/2 = \tau\omega/2 = 1950*53.9/2 = 52.6 kW

This also looks incorrect to me. The average power is not just the work done divided by 2; the formula is something like:

<br /> P_{\rm ave} = \frac{\mbox{work}}{\Delta t}<br />

Your book may have it written differently, of course. But the point is its the fraction (work done in some time interval)/(time interval).

e) Pins = \tau\omega = 1950*53.9 = 10.5 kW/s
 
thank you,
You are correct I did forget to square the angular speed.
But on the average power - it does work both ways. If you note I wasn't taking the work done I was taking the change in Power/2 {(torque*angular speed)-0}/2.

thanks again for your observations, I would probably have lost points on this had I not asked for help.
 
Sorry about part d; I was so focused on the dividing by 2 that I did not actually read what you wrote down, and just assumed you did the (work)/(time) route!