Re: Another Rotating Cylinder Problem - explain from moving frame view
- From: sal <pragmatist@xxxxxxxxxx>
- Date: Wed, 19 Apr 2006 22:16:45 -0400
On Tue, 18 Apr 2006 14:07:07 +0000, David wrote:
On 17 Apr 2006 13:10:24 -0700, "sal" <SpamMeHere@xxxxxxxxxx> wrote:
David wrote:The rotating cylinder problem is easy to visualize. I don't need an
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?
extensive math explanation just some simple verbage.
<g> OK, then let me ask you a simple question about it.
Let's suppose the cylinder lies on the X axis.
So, if the wire runs straight from one disk to the other, we would say
that it makes an angle of 0 degrees with the X axis.
If, on the other hand, it is in a circle around the cylinder -- not a
spiral at all, just a single loop -- then it makes an angle of 90
degrees with the X axis.
_IF_ the wire is to appear _STRAIGHT_ in the moving frame, what's the
_MAXIMUM_ angle it can make with the X axis?
Obviously, it can't cross the X axis at 90 degrees, since in that case
it's just a circle, not a spiral. But can it cross the X axis at
angles arbitrarily close to 90 degrees? In other words, can it form a
really _tight_ spiral? Or is there some limit to the "tightness" of
the spiral it can make, and still appear to be completely straight in
the moving frame?
Please post your answer. :-)
Actually the problem of the spinning spiral is pretty cool, and I
hope to have more to say about it later. The wax cylinder makes it
more graphic but you don't really need it in order to have a very
confusing problem -- the spiral of wire alone will do the job!
Here's the same problem without using rotations.
No, no, let's stick with one problem at a time. Besides, the spiral
problem is quite interesting all by itself, as I already said.
The acceleration on the spiral of wire is the real killer -- the
problem below hasn't got that aspect to it (which certainly makes it
easier to deal with).
This version doesn't have the graphic visualizations of the rotating
cylinder problem. I'll provide the simple physics verbage for the
effect in the rest frame and perhaps you can provide the simple
physics (or math) verbage in the moving frame which I will start.
In the rest frame I have a long rectangular steel rod (like a long
two by four) aligned on the x-axis. The two end surfaces of this
rod are perpendicular to the x-axis. Perpendicular to the x-axis is
a wide conveyer belt (as wide as the rod is long). The belt is
moving with some low speed along the y-axis. Now I have a moving
frame that has velocity V=0.866c along the x-axis with respect to
this rest frame. At time t0 moving frame observesr simultaneously
place all points of the rod on to the conveyer belt. The placement
is parallel to the x-axis.
Rest Frame View
After the accelerations have stopped and the rod is moving at the
same rate as the conveyer belt the rest frame observers note that
the rod is no longer parallel to the x-axis. They say this occurred
because one end of rod was placed on the conveyer belt before the
other end (SR view). They also measure that the end surfaces of the
rod are also no longer perpendicular to the x-axis. They note that
the acceleration of the rod was so small that chemical bonds
remained intact and the rod did not change shape.
Moving Frame View Now what is the verbage the moving frame uses for
the same events? Observers in the moving frame measure that long
edges of the rod remains parallel to the x-axis. But they observe
that the rod has changed shape. The end surfaces of the rod are no
longer perpendicular to the x-axis, nor to the sides of the rod.
What explanation is given by these observers to explain why the rod
changed shape? That is what I'm trying to understand.
While I think about this one, you think about it too, and see if you
can answer this simple question:
Given that the rod appears parallel to the X axis in the "moving"
frame, what is the _maximum_ angle the rod can (appear to) make with
the X axis in the "stationary" frame?
Can the rod appear to be arbitrarily close to an angle of 90 degrees
with the axis, or is there some smaller upper bound on the angle?
It's the same question as in the spiral case, of course, and once
again it points up the fact that just imagining the objects is
insufficient to understand what's going on.
You should also think about ways you could answer _all_ questions
about the rod's orientation and apparent shape. How would you go
about it, if you _had_ to find the answer and nobody was here to work
it out for you? Could you apply the Lorentz transforms in some way?
Please think about that and try to post a procedure for doing it.
--
Nospam becomes physicsinsights to fix the email
I can be also contacted through http://www.physicsinsights.org
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