"The Right Angle Lever Paradox"

The "Right Angle Lever Paradox is a classic construct which is taught
in most courses in Special Relativity. As with all paradoxes, it reveals
that and error has been made in our thinking. It may be interesting then to
examine this paradox and the means that is conventionally used for its
resolution.

The Right Angle Lever paradox reveals itself when we consider a right
angle lever with forces applied to the ends in two different velocity
reference frames. The arrangement is diagrammed as
http://einsteinhoax.com/rf511.gif. In this diagram the lever is shown as
observed in its own reference frame in Figure B and as observed in a
reference frame which is moving at velocity V with respect to the lever in
figure A. The lever is aligned with one of its arms parallel to the velocity
vector between the reference frames and in both reference frames the lever
is observed not to rotate in response to the forces applied to its ends.

In order for the lever not to rotate in response to the forces applied
to the ends of the lever, it is necessary that the torques generated on each
of the arms be equal and opposite, as observed in both reference frames (A
and B). Because of the relativistic contraction observed for the parallel
arm, as observed in reference frame B, the moment applied to the parallel
arm is observed to be reduced by the factor (1-V^2/C^2)^0.5 multiplied by
the Lorentz Transformation for Parallel Force as compared to the moment
observed in reference frame A. In the transverse axis there is no
relativistic shortening of the lever arm and the moment applied to the
transverse arm, as observed in reference frame B is equal to the transverse
force multiplied by the Lorentz Transformation for Transverse Force and it
would seem that, for the lever not to rotate in either reference frame, the
Lorentz Transformation for Transverse Force would have to be (1-V^2/C^2)^0.5
times the Lorentz Transformation for Parallel Force. These transformations
were derived (Minkowski) and, most embarrassingly, the required relationship
was not obtained. The Lorentz
Transformation for Transverse force was found to be the inverse of what was
required to prevent the rotation of the lever, or (1-V^2/C^2)^0.5!

It was obvious early on that the paradox required a further
explanation. Either the derivation of the Parallel and/or Transverse
Transformations for force were faulty or the moments applied to the arms of
the lever did not have to balance in order to prevent rotation. Instead of
accepting that there was a flaw in the derivation(s) of the Parallel and/or
Transverse Transformations, a different and highly creative approach was
taken. It was asserted that, in reference frame B, the force applied to the
end of the parallel lever added energy to it at the rate of Fp*V and added
angular momentum to the lever at the rate of Fp*L. It was then argued that
the rate at which energy was added to the lever and the rate at which
angular momentum was added to the lever produced equal and opposite effects
and the lever did not rotate in either reference frame! IT SHOULD BE NOTED
THAT THE DISCUSSION UP TO THIS POINT IS IN COMPLETE AGREEMENT WITH STANDARD
TEXTS ON THE SUBJECT. From this point on , however, the discussion diverges
from the texts.

If one examines the expression for the angular momentum of an object
one will note that its angular momentum about an axis is the product of the
moment of inertia about that axis and the angular velocity about that axis.
Since the lever is observed not to rotate about its pivot pin axis in either
reference frame, one must conclude that, since its moment of inertia is not
infinite, ITS RATE OF CHANGE OF ANGULAR MOMENTUM MUST BE ZERO as observed in
both reference frames! Next, if one examines any text on basic mechanics one
observes that, in order for a torque to exist, a couple must ales exist. (A
couple is defined by the presence of equal and opposite forces separated by
a distance. The torque is equal to the product of the separation between
these forces and their magnitudes.) In the case of the lever, the couple
results from the presence of the force at the end of the lever and the
resulting reaction force component at the hinge pin which is equal in
magnitude and opposite in direction to the force at the end of the lever.
(This is a requirement of classical mechanics. Advanced physics and cannot
be by-passed by the use of more advanced physics.) When these effects are
considered, the supposedly elegant solution to the Right Angle Lever Paradox
breaks down to the statement that zero=zero. This is most certainly true,
BUT IT IS HARDLY MEANINGFUL.

The Lorentz Transformations for Parallel and Transverse Force are
readily derived without the use of advanced mathematics or Electromagnetic
Theory (apparently used by Minkowski and which has the potential for
introducing error). All that is needed are the well known Lorentz
Transformations of the Special Theory of Relativity, the recognition that
E=M*C^2, and simple algebra. It is readily shown that the Lorentz
Transformation for Parallel Force as currently provided is correct but the
correct value for the Lorentz Transformation for Transverse Force is the
reciprocal of the accepted value. The correct transformation is 1/
(1-V^2/C^2)^)0.5. With this transformation, the right angle lever paradox is
no longer a paradox. What it signified is that the accepted derivation of
the Transformation for Transverse Force was erroneous. Apparently this error
was not recognized because it was inconceivable that a mathematical approach
could produce a faulty conclusion. Lesson:- anyone or anything can screw up.

The material which derives the writer's conclusions is provided at
http://users.isp.com/retic/relcor.htm for your reference. The writer has
received an E-mail from an individual which asserted that he had derived the
Lorentz Transformation for Transverse Force using Maxwell's Equations and
found its accepted value to be correct. He probably used the method used by
Minkowski. That method, since it involves using the velocity of light, would
probably produce the observed error since the velocity of light is must be
considered in both reference frames and velocity is measured using both
length and time. The writer doesn't know the exact nature of his error and
frankly, he doesn't care.

The source material for this posting may be found in
http://einsteinhoax.com/hoax.htm/ (1997);
http://einsteinhoax.com/gravity.htm (1987); and
http://einsteinhoax.com/relcor.htm (1997). EVERYTHING WHICH WE ACCEPT AS
TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE WE HAVE ACCEPTED AS TRUE, IT
MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND IT MUST BE MATHEMATICALLY
VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS REQUIREMENT. THE WORLD IS
ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP FROM THOSE IT HAS GRANTED WORLD
CLASS STATUS.

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Challenges to date have revealed only the responder's inadequacy with one
exception for which a correction was provided.

Dlo wrote:
"The Right Angle Lever Paradox"

The "Right Angle Lever Paradox is a classic construct which is taught
in most courses in Special Relativity. As with all paradoxes, it reveals
that and error has been made in our thinking.

Spammer. In order to avoid spam filters he keeps changing his name
handle - that tells you right away how much he and his "theories" are
worth.

--
Jan Bielawski

"JanPB" wrote in message
oups.com...
| Dlo wrote:
| "The Right Angle Lever Paradox"
|
| The "Right Angle Lever Paradox is a classic construct which is
taught
| in most courses in Special Relativity. As with all paradoxes, it reveals
| that and error has been made in our thinking.
|
| Spammer. In order to avoid spam filters he keeps changing his name
| handle - that tells you right away how much he and his "theories" are
| worth.
|
| --
| Jan Bielawski

Oh, how ****ing observant of you! Well spotted, it's only taken you
two ****ing years to catch on, moron. That tells you right away how
dull-witted and slow you are. And since he'll just ignore you anyway,
you are spamming.
Androcles.

In article .com, JanPB
says...

Dlo wrote:
"The Right Angle Lever Paradox"

The "Right Angle Lever Paradox is a classic construct which is taught
in most courses in Special Relativity. As with all paradoxes, it reveals
that and error has been made in our thinking.

Spammer. In order to avoid spam filters he keeps changing his name
handle - that tells you right away how much he and his "theories" are
worth.

--
Jan Bielawski

The clue, though, is that his post titles are always in double-quotes,
for some reason. Maybe that could serve as the basis for a filter?

--
Daryl McCullough
Ithaca, NY

"Daryl McCullough" wrote in message
...
| In article .com, JanPB
| says...
|
| Dlo wrote:
| "The Right Angle Lever Paradox"
|
| The "Right Angle Lever Paradox is a classic construct which is
taught
| in most courses in Special Relativity. As with all paradoxes, it
reveals
| that and error has been made in our thinking.
|
| Spammer. In order to avoid spam filters he keeps changing his name
| handle - that tells you right away how much he and his "theories" are
| worth.
|
| --
| Jan Bielawski
|
| The clue, though, is that his post titles are always in double-quotes,
| for some reason. Maybe that could serve as the basis for a filter?

Yep... and his name is always a simple word in reverse character order,
as in "Old" this time around. His posts have nothing new in them either.
He's an aetherialist who thinks aether is the only alternative to
relativity,
although most aetherialists are now long gone.
Check out the gif, McCullough, I've used your numbers.
http://www.androcles01.pwp.blueyonde...mart/train.gif
Neat how the clocks run fast when the light is moving to the left, huh?

Androcles

"Dlo" wrote in message
.. .
"The Right Angle Lever Paradox" ...
http://einsteinhoax.com/rf511.gif ...

Actually, the solution to resolve this paradox is ridiculously simple.
If the lever is not rotating in the rest frame, the forces acting on
each level must be identical (only one shown in Retic's diagram). In
the moving frame, both of these forces on each lever would undergo the
same transformation of observation. Since both of these forces for
each lever would undergo the same transformation, the net force is zero
as well. The other level does not have to go through the same
transformation, bu the two forces acting on that level have to go
through the same transformation relative to each. The two forces in
opposite directions are always the same in each lever through any
transformations. There is no net force on each lever, and thus the
lever should not rotate in any frame of reference. This is not a
paradox.

However, the Twin's Paradox is still a paradox that needs to be
resolved.

"Koobee Wublee" wrote in message
oups.com...
"Dlo" wrote in message
.. .
"The Right Angle Lever Paradox" ...
http://einsteinhoax.com/rf511.gif ...

Actually, the solution to resolve this paradox is ridiculously simple.
If the lever is not rotating in the rest frame, the forces acting on
each level must be identical (only one shown in Retic's diagram). In
the moving frame, both of these forces on each lever would undergo the
same transformation of observation.

There is only one lever. Read again. ;-)

Harald

Since both of these forces for
each lever would undergo the same transformation, the net force is zero
as well. The other level does not have to go through the same
transformation, bu the two forces acting on that level have to go
through the same transformation relative to each. The two forces in
opposite directions are always the same in each lever through any
transformations. There is no net force on each lever, and thus the
lever should not rotate in any frame of reference. This is not a
paradox.

However, the Twin's Paradox is still a paradox that needs to be
resolved.

Koobee Wublee:
"Dlo" wrote in message
. ..
"The Right Angle Lever Paradox" ...
http://einsteinhoax.com/rf511.gif ...

Actually, the solution to resolve this paradox is ridiculously simple.
If the lever is not rotating in the rest frame, the forces acting on
each level must be identical (only one shown in Retic's diagram). In
the moving frame, both of these forces on each lever would undergo the
same transformation of observation.

Wrong. That is not the resolution. Forces perpendicular to and
parallel to the transformation transform differently. Try again.

[...]
lever should not rotate in any frame of reference. This is not a
paradox.

Well, it certainly has a resolution, although you haven't found it.

However, the Twin's Paradox is still a paradox that needs to be
resolved.

Only because you don't understand its resolution, as is evident
from your attempt to resolve the one above.

Bilge wrote:
Koobee Wublee:
Actually, the solution to resolve this paradox is ridiculously simple.
If the lever is not rotating in the rest frame, the forces acting on
each level must be identical (only one shown in Retic's diagram). In
the moving frame, both of these forces on each lever would undergo the
same transformation of observation.

Wrong. That is not the resolution.

I interpreted Wublee's response such that it is indeed correct, albeit
poorly stated: consider a small portion of one part of either lever --
the total force is zero in the rest frame (because it is not moving), so
the external force and the internal forces (from neighboring portions of
the lever) must therefore cancel; that means they are equal and
opposite, and since anti-parallel forces transform the same, the same
conclusion must hold in the moving frame.

The poor part of his statement is "the forces" -- _which_ forces???


Forces perpendicular to and
parallel to the transformation transform differently.

This is true, but given my interpretation of his writing it is not relevant.


However, the Twin's Paradox is still a paradox that needs to be
resolved.

Only because you don't understand its resolution,

Agreed.


Tom Roberts

Tom Roberts:
Bilge wrote:
Koobee Wublee:
Actually, the solution to resolve this paradox is ridiculously simple.
If the lever is not rotating in the rest frame, the forces acting on
each level must be identical (only one shown in Retic's diagram). In
the moving frame, both of these forces on each lever would undergo the
same transformation of observation.

Wrong. That is not the resolution.

I interpreted Wublee's response such that it is indeed correct, albeit
poorly stated: consider a small portion of one part of either lever --
the total force is zero in the rest frame (because it is not moving), so
the external force and the internal forces (from neighboring portions of
the lever) must therefore cancel; that means they are equal and
opposite, and since anti-parallel forces transform the same, the same
conclusion must hold in the moving frame.

You are presuming the result under the (unstated) premise of absolute
simultaneity. If the ``frame of the lever'' is defined such that every
point on the lever has a spacelike separation, the forces at different
points along the lever aren't even relevant to the question[1]. The
only forces that are relevant are the ones at the event defined to be
the point of rotation.

If you choose the pivot point for that event, then the result is simple.
There is no rotation about the pivot, so the torques at the pivot must
be zero. Choosing the lever arms as two of the spatial axes with the
forces applied at some distance perpendicular to those axes, call them
x and y, the torque in the pivot rest frame must be N = x F_y - y F_x = 0.
Choosing the origins to be coincident, under a boost in the x-direction,
F'_x = F, F'_y = \gamma^-1 F_y and the distance (x'-0) = \gamma^-1 (x - 0),
(y' - 0). Hence the forces in the primed frame are _not_ equal, even
though the torques are.



[1] Alternatively, you could to define frames such that he points that
define the lever frame are causally related, but that requires using
one of those funny frames which is poincare invariant (and relativ-
istically correct) but is not related to the standard coordinates by
a lorentz transform. Since just about everyone objects to such
coordinates for reasons that I've noted before, I assume no one is
referring to such coordinates.


The poor part of his statement is "the forces" -- _which_ forces???

The only forces that matter are the ones at the event defined as the
point about which the rotation is being considered.