Rotational Kinematics? Revolutions of a propeller?

[JHCreighton]

Homework Statement

An airplane engine starts from rest; and 2 seconds later, it is rotating with an angular speed of 420 rev/min. If the angular acceleration is constant, how many reovlutions does the propeller undergo during this time?

Homework Equations

$$\theta$$ = $$\omega$$ ₀ t +1/2 $$\alpha$$ t^2

$$\alpha$$ = $$\Delta$$ $$\omega$$ / $$\Delta$$

The Attempt at a Solution

I thought that if by converting rev/s to rad/s I could use the angular acceleration to somehow find the number of revolutions made in 2 seconds. But I'm not sure if this requires converting, or even if I am on the right track with the angular acceleration.

 

[Ush]

1. to do your calculations, you do require converting from rev/m to rad/s
2. what equation relates angular velocity to angular acceleration and time?

 

[JHCreighton]

1) So, 420 rev/s would be 840(pi). (I can't find the Pi character) because 1 rev=2(pi) rad. Correct?

2) The equation should be $$\omega$$ = $$\omega$$ ₀ + $$\alpha$$ t

 

[Ush]

1) So, 420 rev/s would be 840(pi). (I can't find the Pi character) because 1 rev=2(pi) rad. Correct?

Your question says 420 rev/m (revolutions per minute), not 420rev/s.
you first need to convert 420 rev/m to rev/s

2) The equation should be $$\omega$$ = $$\omega$$ ₀ + $$\alpha$$ t

correct,
you are given enough information to calculate angular acceleration.

now, you can use your angular displacement equation to calculate the number of radians rotated in 2 seconds.

you can then convert radians to revolutions.

 

[JHCreighton]

OK, so let's see. 420 rev/MIN is 7 rev/s, which is 44 rad/s. Rearranging the kinematic equation to be $$\alpha$$ = v_f /t and substituting, I get the angular acceleration to be 22 rad/s^2. Then, if the initial velocity is 0 rad/s, the final velocity is again 44 rad/s. This converted to rev/s is 44/2(pi), which equals 7 revolutions. I think I took the long way around to get here, but does this seem like the correct process? Or at least it lead to the correct answer?

 

[Ush]

after you find your angular acceleration, (22rad/s^2). You need to find your angular displacement. (Remember, angular displacement is the number of radians the propeller turns)

so no, 7 revolutions is not the correct answer.

 

[JHCreighton]

Jeez, this problem has just worked me over, and for no good reason. OK, simply substituting everything into the angular displacement equation, then converting back to revolutions, I get 14 rev. I can't tell you how much I appreciate the help, especially since you took the time to go step by step with me, without just tossing out everything I needed at once.
My sincerest thanks,
JHCreighton

 

[Ush]

sorry, 7 revolutions was the correct answer. I didn't do the math, I just looked at what you were saying.. But if you do the math
--
wf^2 = wi^2 + 2⍺θ
rearrange for θ

OR

θf = θi + Wit + ½⍺t^2
rearrange for θ
--

θ turns out to be 44radians = 7 revolutions =p