How Does Temperature Affect the Angular Velocity of a Rotating Aluminum Wheel?

[darkAzNk3n]

Homework Statement

A 28.4-kg solid aluminum cylindrical wheel of radius 0.41m is rotating about its axle in frictionless bearings with angular velocity w = 32.8 rad/s. If the temperature is then raised from 20.0 degrees celcius to 95.0 degrees celcuis, what is the fractional change in angular velocity w?

Homework Equations

delta l = alpha*l*delta T T=temperature
v=rw w=angular frequency

The Attempt at a Solution

I used the thermal expansion equation and found the change in Radius r.
I also assumed that because the wheel was on frictionless bearings that velocity v was constant.
so finding the change in r, i pluged into the angular frequency equation delta w = v/delta r.

is that in any way correct?

 

[diazona]

I also assumed that because the wheel was on frictionless bearings that velocity v was constant.

This assumption is wrong, but other than that, you're kind of thinking along the right lines.

Think about it this way: the fact that the wheel is on frictionless bearings means that the force acting to rotate it is zero. That in turn means that the torque on the wheel is zero. What quantity is constant (a.k.a. conserved) for a rotating object with no torque on it?

 

[darkAzNk3n]

. What quantity is constant (a.k.a. conserved) for a rotating object with no torque on it?

the only thing i can come up with is that the radius is constant but, wouldn't the wheel have to expand which would increase the radius for there to be a change in the angular velocity?

 

[diazona]

Nope, it's not the radius.

What conservation laws have you learned about?

 

[darkAzNk3n]

Nope, it's not the radius.

What conservation laws have you learned about?

Conservation of energy and momentum pop up at the top of my head

 

[diazona]

OK, good, but there's a third conservation law that you'll need for this problem. What's the one that applies specifically to rotating objects?

 

[darkAzNk3n]

OK, good, but there's a third conservation law that you'll need for this problem. What's the one that applies specifically to rotating objects?

Conservation of Angular Momentum?

dL/dt=0 and L = Iw = constant

I think.

 

[diazona]

Yep, that's the one. Now how can you use that in this problem?

As a first step, what's I for a solid cylindrical wheel?

 

[darkAzNk3n]

I got I = 1/2 mr^2

then L = 1/2 mwr^2

but what bout the L?

do i find the change in radius with the thermal expansion equation then plug that radius back into the angular momentum equation?

 

[diazona]

Why, do you have another idea as to how to proceed?

Try it and post your work here

 

[darkAzNk3n]

hmm

since we know L=Iw and since I=Mr^2/2, L = (wMr^2)/2=constant

and from thermal expansion equation, we know delta r = alpha*r0*delta T

we can then add r0 and delta r to find the new r

then we plug back into L = (wMr^2)/2
and from the conservation of angular momentum we know that L is a constant, we can find new w with the new r, w= 2L/Mr^2
and compare with initial w that's given

hard to follow sorry.

 

[diazona]

Yep, that's exactly the way to do it.

 

[darkAzNk3n]

Yep, that's exactly the way to do it.

o Thanks for the help haha
one down.. four to go.

 

[darkAzNk3n]

for the fractional change of w, I got something small like 3.74 * 10^-3 rad/sec. is that right?

 

[diazona]

Assuming you did the math correctly, I don't see why it shouldn't be.