Difficult problem on Angular Kinetics

[Strontium90]

Hi, I was working on a problem and I am having trouble in being able to solve this problem. The difficulty is that the question only presents not enough data to find the answers that it asks for.

Homework Statement

Here is the problem in it's entirety:

A wheel whose moment of inertia is 0.4 slug.ft^2 is rotating at 1500 rev/min. (a) what constant torque is required to increase its angular velocity to 2000 rev/min? (b) How many turns does the wheel make while it is being accelerated? (c) How much work is done on the wheel?


the answers from the book are:
(a) 2.63 lb.ft
(b) 234 turns
(c) 3.86 X 10^3 ft.lb

To answer this question, you need to find what torque is used to increase the angular velocity. As far as I can tell, you cannot do that because you do not have the time it took to increase the velocity, the displacement for the increase of the velocity or the angular acceleration.

Homework Equations

The equations that I used are:

θ = ωt + 1/2αt^2

ω = ω + αt

ω^2 = ω^2 + 2αθ

τ = Iα

The Attempt at a Solution

The attempts that I made to find a solution involved trying to find the displacement from the acceleration of the wheel. This course of action did not lead to the answer that the book had. My best guess was the second equation, the equation for velocity, acceleration and displacement. I tried to make a guess of the amount of displacement because time, acceleration and displacement were not available.

The only information was the moment of inertia and the initial and final angular velocities of the wheel. My goal is to find a way to get information out of a problem of this nature and have a method to do this in the future. Thank you.

 

[TSny]

You're right. Without the time you can't get a numerical answer for (a) or (b). You should be able to get an answer for (c).

 

[Strontium90]

But the book has an answer for (a) of this problem. Is there some way to deduce the torque needed to accelerate the wheel? When I looked at this problem, I suspected that without the time, acceleration or displacement to the new angular velocity, there would be no way to determine the torque. Hypothetically you could have any torque, as long as it is greater than 0.4 lb.ft.

 

[TSny]

Yes, you should be able to work back from the answer for (a) or the answer for (b) to find the time.

Actually, any torque greater than 0 would eventually bring the angular speed up to 2000 rev/min.

 

[Strontium90]

Hmmm, wouldn't that be counterproductive? Isn't there a way to find the acceleration without the time and with the initial and final velocities?

 

[TSny]

No. Without the time there is no way to determine the torque or the angular acceleration. You could apply a very small torque for a long time and you will eventually get the final speed up to 2000 rev/min. Or you could apply a large torque for a small time and get the speed to 2000 rev/min. The wheel will make many more turns for the case of the small torque compared to the large torque.

 

[D H]

Hmmm, wouldn't that be counterproductive? Isn't there a way to find the acceleration without the time and with the initial and final velocities?

No, there isn't. Think of it in terms the linear analog of this problem. What constant force is needed to make a 4000 pound car accelerate from 30 mph to 60 mph? Any positive value will do, given enough time. This question is unanswerable, and so is the question in your text. Not enough information was given. Textbooks occasionally have errors. This is one of them.

Going back to the original post,

The only information was the moment of inertia and the initial and final angular velocities of the wheel. My goal is to find a way to get information out of a problem of this nature and have a method to do this in the future.

How to solve problems like these? You can't. There's missing information that makes the question unanswerable. Check if there's an errata sheet for your text. The needed information might be there. If not, ask your professor about the missing information or write the author of the text.

This is typical of real world problems. The needed information isn't always there. You have to fill in those blanks somehow, but the first step is to recognize that some crucial piece of information is missing.