Re: Simple length problem - for Todd, Harald, thinkers
- From: David <dseppala@xxxxxxxxxxxxx>
- Date: Mon, 03 Jul 2006 01:35:18 GMT
On Sun, 2 Jul 2006 16:42:33 -0500, "Todd" <nospam@xxxxxxxxxx> wrote:
I am suggesting that there is some range of stretching where indeed
"David" <dseppala@xxxxxxxxxxxxx> wrote in message
news:qf6ga21rq2958niftna17dtq7nkbmtvh4n@xxxxxxxxxx
Todd, lets look at and agree on some basic properties of materials. I
want to make sure I understand the simple cases as explained in SR. In
SR problems instead of using high velocities and everyday distances, I
look at things using everyday velocities and very, very long lengths
(this is what Dirk calls my immortal fumble).
Let's say that a rod of a given material, length and diameter can be
stretched a bit and when the force that causes the stretching is
removed the rod returns to its original length. Posters in this group
have indicated that speed the rod will return to its original length
is the speed of sound for the material. Let's assume that is correct.
Let's agree that there is some length such that if the rod is
stretched beyond this critical length, the rod will not return to its
original length when the force is removed. The rod will remain at a
length longer than this critical length due to changes in chemical
bonds. This is a well known behavior of materials.
Now, let's say we have a rod of diameter D composed of N atoms. Lets
say this rod is extremely long and there is a line of people holding
this rod along the x-axis. Let's say the rod is stretched to its
critical length. Call that length L. Lets say there is a conveyer
belt moving with velocity V along the x-axis that has an identical rod
of N atoms on it that is also stretched to its critical length.
Now at time t0, when one end of the rod on the conveyer belt is at the
same point in space as one end of the rod the people are holding,
everyone simultaneously steps onto the moving conveyer belt while
holding the rod. This rod accelerates to velocity V. According to
SR, this rod is now longer than the identical conveyer belt rod. The
rest frame observers said the conveyer belt was shorter than the rest
frame rod, and when eveyone stepped onto the conveyer belt this very
long rod didn't suddenly contract to the same length as the conveyer
belt rod (since that occurs at the speed of sound and for rods 100
light-years in length that would take tens of thousands of years).
Observers on the conveyer belt said that the rod started out shorter
but was stretched when everyone stepped onto the conveyer belt because
one end stepped onto the conveyer belt before theo ther end.
So observers in both frames agree that the original rest frame rod is
now stretched beyond its critical elastic limit and cannot be returned
to its original length. So now let's say one nano-second after time
t0, all of these people holding the rod simultaneously, as measured by
their clocks (the original rest frame times) step back into their
original frame while holding still holding this rod. Now this rod is
the same length that it originally was since everyone only traveled a
distance of one-nano second time V (I used V = 3 meters/second).
But now we have a conflict with what we agreed on. We agreed that the
rod was initially stretched to its critical length L and any further
stretching would cause it to remain stretched beyond L. We could not
do anything to return the rod to its original length if it was
stretched beyond this critical length.
Why can't you apply compressional forces to bring the rod back to its
original length?
you can bring the rod back to its original length. But I am also
suggesting that there is some critical length where stretching beyond
this length changes the chemical bonds and that compression will not
restore it to its original length. We can take a steel spring,
stretch it to its critical length and release the force causing the
stretching and the spring will return to its natural unstretched
length. However there is some critical length we can stretch the
spring beyond and when the stretching force disappears the spring will
not return to its original length. You can try to reverse the steps,
but the stretched spring won't return to its original shape. The same
thing happens if it is a steel rod and not spring shaped. There is a
critical length beyond which the rod will not return to its original
length with mere compression. The chemical bonds have been changed
and you need more than just compression to reverse the process. Some
materials will fracture if they are stretched too far. Trying to
compress the materal to reverse the process will not remove the
fracture.
So when everyone steps onto the conveyer belt with some materials,
the rod will simply fracture, and stepping off the conveyer belt back
into the original frame will not make that fracture disappear.
Likewise, when the material is something that doesn't fracture, but is
stretched beyond its critical limit, it will not return to its
original length. In the steel rod case the shape will change. The
diameter of the rod will be reduced as the length is stretched. There
is some point in the stretching where the chemical bonds change enough
that compression won't return the rod to its original diameter.
There are many chemical properties which once done cannot be undone
by reversing the same steps. Stretching of materials past their
elastic limit is one such property.
Look at Einstein's theory using a rod that is a few hundred light
years long and use a very low velocity like 3 meters / second as the
difference in the two frames.
From the original rest frame view, when everyone steps on to the
conveyer belt simultaneously as measured in the rest frame, they say
that in just a fraction of a second all the chemical bonds increase
their spacing along the x-axis by some small amount. This occurs in
Einstein's theory because the length of the rod can't instaneously
change (which is easy to show) but the atoms must be smaller due to
their velocity V. The length contraction of individual atoms occurs
over the width of each atom and can travel that distance very quickly.
Let's say that happens at either the speed of sound or the speed of
light. But since the rod is over 100 light-years in length, when
everyone simultaneously steps onto the conveyer belt, the rod that
everyone is holding is now 3 or so meters longer than the identical
rod that has always been in the conveyer belt frame. The size of the
individual atoms change almost instaneously, but we have to wait at
least 100 years for the length to change (if things travel at the
speed of light) to be equal in length of the non-accelerated rod, or
maybe tens of thousands of years if the effect occurs at the speed of
sound (as changes in chemical bonds do). So observers in this
original rest frame see the chemical bonds stretched almost
instaneously.
What do observers in the conveyer belt see? They see one end of
the rod accelerated (decelerated) before the other end - say with the
lengths and speeds this time difference is one second. They say the
rod had been shorter than their 100+ light-year rod before everyone
stepped on the conveyer belt. And now its a meter longer than their
identical 100+ light-year rod. They say the individual atoms have
increased in size, and since this was over a very small distance for
each atom, this happened almost instantaneously for each individual
atom, so within a second or so all the atoms increased in size. And
within a second or so the length of the rod stretched. But to have
the rod increase in length from being shorter than their rod to being
longer than their rod, this effect has to propagate through all the
chemical bonds. Somehow, these observers have to measure that some
effect relating to the chemical bonds was able to propagate over a
length of 100+lightyears in just a second or two. That cannot happen.
David
.
In stepping onto the conveyer
belt we stretched the rod beyond its critical length. All frames
agree that this occurred. But when we stepped back, the N atoms have
a length of L - they are no longer stretched beyond this critical
length. In other words, the rod somehow (if you use SR concepts) was
stretched beyond its elastic limit, yet returned to its original
length by a mere application of an acceleration which changed its
velocity by 3 meters/second (using the value I used for the velocity
parameter). I don't see how SR notions of length and time work in
this simple case.
The "mere" acceleration that returns the rod to the rest frame requires
compressional forces to be applied to the rod and that is what brings the
rod back to its original length.
If you could add some insights here, then perhaps I can make some
progress on the magnet, cylinder problems.
Thanks,
David
When going from the rest frame to the belt frame, the proper length of the
rod increases. If Lo is initial proper length, then the new proper length
will be g*Lo, where g is the gamma factor relating the two frames. The
fractional change in length is then (g-1). For v = 3 m/s, the fractional
change in length is something like 5 x 10^(-17). Now it's the fractional
change in length that determines the stress built up in the rod. So, it's a
bit hard to imagine being so close to the elastic limit that an additional
fractional change of length of only 5 x 10^(-17) takes the rod beyond the
elastic limit! But this is a thought experiment, so let's go with it.
When you say that the rod is stretched beyond its elastic limit, you just
mean that the rod will remain stretched if no external forces are applied to
the rod. It doesn't mean that you can't compress the rod back to its
original length if sufficient external compressional forces are applied.
When the rod-holders step from the belt back to the original rest frame,
there will be compressional forces bringing the rod back to its original
length. Once back in the rest frame, the holders might need to keep
applying a compressional force if they want the rod to remain at its
original length. If they let go of the rod once its back at rest relative
to the original rest frame, then the rod might relax to a longer length -
depending on the material properties of the rod.
Suppose you replace the rod by just two blocks connected by a spring laid
out at rest along the x-axis of the rest frame. The spring is assumed to be
stretched almost to its elastic limit. (Forces would have to be applied to
the blocks to overcome the force of the spring on the blocks so that the
blocks can remain at rest.) Then additional forces are applied to the
blocks simultaneously from the point of view of the rest frame to give the
blocks identical accelerations until the blocks have a speed V relative to
the rest frame along the x. In the comoving frame of the blocks, the
distance between the blocks has increased. (This is just the famous "Bell
spaceship paradox".) Thus the spring is now stretched beyond its elastic
limit. But that doesn't mean that you can't compress the spring back to it's
previous length if you apply forces to it.
When forces are applied to the blocks to decelerate them (simultaneously
again with respect to the rest frame) back to being at rest in the rest
frame, the spring will return to the same length it had at the start of the
experiment. But, now the spring doesn't necessarily exert the same force on
the blocks as it did at the start of the experiment because the spring was
stretched beyond its elastic limit during the experiment. Where's the
problem?
Todd
(PS Today is my last day of vacation so, unfortunately, I will not be able
to post as often.)
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