Re: Another Rotating Cylinder Problem - explain from moving frame view
- From: "sal" <SpamMeHere@xxxxxxxxxx>
- Date: 17 Apr 2006 13:10:24 -0700
David wrote:
Can anyone explain this rotating disk problem from the point of view
of a moving observer?
In the rest frame let there be two rotating disks of diameter D
perpendicular to the x axis. Let the distance between the disks be
L. Let there be a rotating cylinder of the same diameter and length
connecting these two disk. Let the disks be massive and made out of
steel and let the cylinder be made out of wax. Let the cylinder and
disks rotate at one revolution per second.
Let there be a frame moving along the x axis relative to this rest
frame with some V. Let L and V be such that simultaneous events
measured in the moving frame at each disk (separation L) are
measured as a half- second time interval in the rest frame. At time
t0 as measured in the moving frame a thin straight wire is
simultaneously attached to the two disks at the top position of each
disk and along the top of the wax cylinder. This is a straight line
in the moving frame, but spirals around the cylinder making a half
revolution as viewed in the rest frame.
Now very slowly the tension of this wire is increased - the wire is
stretched. This means the wire is very slowly approaching a
straight line as viewed in the rest frame. As the tension is
increased this wire cuts through the wax cylinder. Eventually the
wire becomes a straight line and any further stretching of the wire
does not change its shape.
As viewed in the moving frame the wire is a straight wire on the
surface of the cylinder rotating with the cylinder before we start
stretching the wire. Now as the wire is stretched the center point
of this wire eventually touches the center of the rotating cylinder
(the x-axis) as the wire slices through the wax. Can anyone explain
as viewed in the moving frame why the center of this straight wire
cuts the wax all the way through to x-axis as the wire is stretched?
You may not realize just how complex this problem is.
You are asking about the forces on and tension in a wire which is in
motion, where the forces and tension are measured by a _stationary_
observer.
For starters, you need to at least think about how a stationary
observer would even _measure_ the tension in a wire that's in motion.
The measurement is surely not going to be the same as the value
measured in the wire's rest frame -- but we need to go beyond that
simple assertion of what it _won't_ be. What _will_ the tension be,
as viewed from the stationary frame? And what does it even _mean_?
Until you've determined how you can measure the tension within an
object which is in motion, without having the measurement apparatus
co-move with the object, you don't have a working definition for
"tension" in a moving object.
The rotating lever paradox is difficult, and it involves just a couple
of torques; what you've described here is even more complex than that.
So I would suggest backing up and taking a running start. Go back to
a simpler problem, and work out an analytic solution to that. Then
tackle the more complex problem.
As a general rule, what you need to do is begin by analyzing the
problem completely in the most convenient frame-of-reference you can
find. Typically, that's the center of mass frame.
Here's an example of a simpler problem, similar to something you've
posted in the past; I'll just sketch it (I'm sure you can fill in the
details): Start with a spinning rod with a (straight) stripe on it.
Viewed in a moving frame the stripe looks like a spiral. In the
center of mass frame, let the stripe "fall off" and fly away. Now,
describe _exactly_ what appears to happen in the moving frame.
Post the answer. Working that through completely will help a lot with
your later problems, I think.
Then take the problem you've posted here, but instead of pulling the
wire through the wax, release it so it can fly off. Figure out what
happens in the center of mass frame (this may prove harder than you
expect, even though it's "just" Newtonian mechanics!). Then map that
into the frame in which the cylinder is moving, and tell us what the
moving observer would see.
Finally, figure out how to transform _tension_ and _force_ between
frames. Figure out the tension and 4-force on the taught wire in the
center of mass frame, and transform that to the moving frame. Tell us
what you found: post the transformation equations.
The latter is going to be difficult but would be a very useful
exercise for you to do, and is a prerequisite for solving the problem
you actually posted.
Once you've done that, you can apply your transformations to the
"straight" wire in the moving frame and see if you can correctly
predict that it will cut through the wax.
I, personally, have no plan to do this for you :-) but I'd be more
than happy to look at (and, if possible, help with) any attempt at a
solution you can come up with.
If you post an explanation, does the same explanation work when the
straight wire is simultaneously attached as measured in the rest
frame and then the wire is slowly stretched?
After you post your explanation, we'll be glad to determine if it so
applies.
That is, the wire is
stretched and stretched but it never cuts into the wax eventhough it
spirals around the cylinder as viewed in the moving frame.
This physics result of SR seems non-sensical to me.
Lots of results in SR seem nonsensical. Unfortunately for our "common
sense" every such prediction which has been tested has been born out.
God apparently doesn't care tuppence for whether reality behaves
according to your -- or my -- common sense.
Thanks,
David Seppala
.
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