The Right Angle Lever Paradox

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

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 in the Figure shown in
. 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
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 "Gravity" (1987),
"The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
Relativity (1999) located at
.
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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E-mail:-

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Einsteinhoax:
The Right Angle Lever Paradox

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

That would explain your opposition to physics.

............. ...That, if a straight line falling on two straight
lines makes the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on which
the angles are less than two right angles!!!!!!!!!!.......... ...

-- Euclid,

--
Ahmed Ouahi, Architect
Best Regards!


"Bilge" kirjoitti viestissä
...
Einsteinhoax:
The Right Angle Lever Paradox

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

That would explain your opposition to physics.

"Einsteinhoax" wrote in message
...
The Right Angle Lever Paradox

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

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.


[SNIP]

The "paradox" you present is indeed an interesting case. Having not
previously come across it, it caught my attention and I spent considerable
time seeking an answer.



First I examined the question of how forces transform under a Lorentz
transformation between inertial frames. I did this by examining the
behavior of an accelerating body (actually a point unit-mass body under
constant acceleration as perceived on board the body), observed both in a
rest frame (the frame in which the body is at rest prior to acceleration)
and in an inertial "moving" frame, related by the usual Lorentz
transformation equations.



I computed the usual kinematic parameter set (time, position, velocity and
acceleration) for each frame's observer. This allowed me to also compute
the momentum-per-unit- mass in each frame as well as the time derivative of
that momentum.



The latter is, by definition, the force-per-unit-mass that produces that
acceleration.



The results clearly show that although observers don't agree about
position,velocity and acceleration, observers in all inertial frames will
agree that the force (the thrust being used aboard the body to accelerate
that body) has the same constant value.



Conclusion: Force (or at least this type of mechanical force) is an
invariant under a Lorentz transformation. [This may indeed be true for
total EM force as well, though the electric and magnetic components vary
between inertial frames.]



But then, why isn't there an unbalanced torque on the bracket in the moving
frame, since clearly the bracket leg along the frame velocity is
foreshortened from its proper length?



The answer is the following, and strangely, it follows from our
understanding of relativistic stellar aberration.



To the observer in the "moving" frame the angle between the two bracket legs
is not a right angle, as it is in the "rest" frame. Think of it this way:
If the "at rest" observer were stationed on the bracket hub, sighting his
telescope along the transverse leg and saw star 'A' (a coincidence which
comprises an event in spacetime), the "moving" observer, sighting on star 'A'
(and therefore also along that leg's length), would find his telescope to be
directed at a different angle relative to the direction of motion.



If PHI is the angle between velocity direction and star in the "rest" frame,
and "phi" in the "moving" frame, then



tan(phi/2) = sqrt{[1-(v/c)]/[1+(v/c)]}*tan(PHI/2).



[This equation results from an application of the Lorentz transformation
equations.]

In the "rest" frame, PHI = 90 deg so that phi = 2* arctan(
sqrt{[1-(v/c)]/[1+(v/c)]}).

The leg rotation is (90 - phi) deg so that the transverse torque lever arm
is F*L*cos(90-phi).

If you do the computation you'll discover that cos(90-phi) = sqrt[1-(v/c)^2]
for all v.



This means that in the "moving" frame the transverse bracket leg is
"rotated" by an angle such that the leg's projection on the "moving" frame's
y-axis is the same length as the other leg's foreshortening.



The zero torque in the "rest" frame remains a zero torque in the "moving"
frame.



Thanks for presenting this question. I learned a lot and it was much fun.

Eli Botkin

Hello again:
I've been continuing to think about this Right Angle Lever Paradox and now
feel that I stated something incorrectly. I believe the conclusion to be
correct. The zero torque in the "rest" frame remains a zero torque in the
"moving" frame. But I had also said that to the observer in the "moving"
frame the angle between the two bracket legs is not a right angle, as it is
in the "rest" frame. That's not right.

The bracket does remain a right angle bracket in the "moving" frame. Rather
it is the force that changes direction by (90-phi) deg relative to the
bracket arm, at the point of application. In the "moving" frame the
direction of the force remains perpendicular to the line-of-sight to star
'A', as it is in the "rest" frame.

The force direction on the y-arm (motion being in the x-direction) no longer
makes a right angle with the y-arm. Its produced torque is now equal but
opposite to the torque from the contracted x-arm. The total torque is still
zero and no bracket rotation is induced. Only the force reactions at the
bracket hinge are changed, a non-paradoxical outcome.

I'm still grappling with finding the SR principle that dictates all this.
[Note that in a sense this would make the force direction (along with its
magnitude) invariant in the absolute spacetime manifold.]

Not yet being able to prove this force-direction change idea, I admit that
it attracts me because of my faith in SR's description of kinematic and
dynamic behavior. It provides a non-paradoxical answer to the problem you
pose and doesn't violate any known mechanics principles. (At least known to
me.)

Thanks again for the entertaining moments spent.
Eli Botkin


"Eli Botkin" wrote in message
news:1111709918.4961a481db139edb244d3e479b4c2c3f@t eranews...
[SNIP]

The "paradox" you present is indeed an interesting case. Having not
previously come across it, it caught my attention and I spent considerable
time seeking an answer.



First I examined the question of how forces transform under a Lorentz
transformation between inertial frames. I did this by examining the
behavior of an accelerating body (actually a point unit-mass body under
constant acceleration as perceived on board the body), observed both in a
rest frame (the frame in which the body is at rest prior to acceleration)
and in an inertial "moving" frame, related by the usual Lorentz
transformation equations.



I computed the usual kinematic parameter set (time, position, velocity and
acceleration) for each frame's observer. This allowed me to also compute
the momentum-per-unit- mass in each frame as well as the time derivative
of that momentum.



The latter is, by definition, the force-per-unit-mass that produces that
acceleration.



The results clearly show that although observers don't agree about
position,velocity and acceleration, observers in all inertial frames will
agree that the force (the thrust being used aboard the body to accelerate
that body) has the same constant value.



Conclusion: Force (or at least this type of mechanical force) is an
invariant under a Lorentz transformation. [This may indeed be true for
total EM force as well, though the electric and magnetic components vary
between inertial frames.]



But then, why isn't there an unbalanced torque on the bracket in the
moving frame, since clearly the bracket leg along the frame velocity is
foreshortened from its proper length?



The answer is the following, and strangely, it follows from our
understanding of relativistic stellar aberration.



To the observer in the "moving" frame the angle between the two bracket
legs is not a right angle, as it is in the "rest" frame. Think of it this
way: If the "at rest" observer were stationed on the bracket hub, sighting
his telescope along the transverse leg and saw star 'A' (a coincidence
which comprises an event in spacetime), the "moving" observer, sighting on
star 'A' (and therefore also along that leg's length), would find his
telescope to be directed at a different angle relative to the direction
of motion.



If PHI is the angle between velocity direction and star in the "rest"
frame, and "phi" in the "moving" frame, then



tan(phi/2) = sqrt{[1-(v/c)]/[1+(v/c)]}*tan(PHI/2).



[This equation results from an application of the Lorentz transformation
equations.]

In the "rest" frame, PHI = 90 deg so that phi = 2* arctan(
sqrt{[1-(v/c)]/[1+(v/c)]}).

The leg rotation is (90 - phi) deg so that the transverse torque lever arm
is F*L*cos(90-phi).

If you do the computation you'll discover that cos(90-phi) =
sqrt[1-(v/c)^2] for all v.



This means that in the "moving" frame the transverse bracket leg is
"rotated" by an angle such that the leg's projection on the "moving"
frame's y-axis is the same length as the other leg's foreshortening.



The zero torque in the "rest" frame remains a zero torque in the "moving"
frame.



Thanks for presenting this question. I learned a lot and it was much fun.

Eli Botkin

Hi once more"
The more I think about this, the more I think I may be barking up the wrong
tree.
Maybe I was resorting to a lot of wishful thinking. If I've wasted too much
of your time, I'm sorry to have done so. I'll certainly keep thinking on
this.
Eli Botkin

"Eli Botkin" wrote in message
news:1111784662.2f90d9f70ab6e95c5c1e72062e8f9165@t eranews...
Hello again:
I've been continuing to think about this Right Angle Lever Paradox and now
feel that I stated something incorrectly. I believe the conclusion to be
correct. The zero torque in the "rest" frame remains a zero torque in the
"moving" frame. But I had also said that to the observer in the "moving"
frame the angle between the two bracket legs is not a right angle, as it
is in the "rest" frame. That's not right.

The bracket does remain a right angle bracket in the "moving" frame.
Rather it is the force that changes direction by (90-phi) deg relative to
the bracket arm, at the point of application. In the "moving" frame the
direction of the force remains perpendicular to the line-of-sight to star
'A', as it is in the "rest" frame.

The force direction on the y-arm (motion being in the x-direction) no
longer makes a right angle with the y-arm. Its produced torque is now
equal but opposite to the torque from the contracted x-arm. The total
torque is still zero and no bracket rotation is induced. Only the force
reactions at the bracket hinge are changed, a non-paradoxical outcome.

I'm still grappling with finding the SR principle that dictates all this.
[Note that in a sense this would make the force direction (along with its
magnitude) invariant in the absolute spacetime manifold.]

Not yet being able to prove this force-direction change idea, I admit that
it attracts me because of my faith in SR's description of kinematic and
dynamic behavior. It provides a non-paradoxical answer to the problem you
pose and doesn't violate any known mechanics principles. (At least known
to me.)

Thanks again for the entertaining moments spent.
Eli Botkin


"Eli Botkin" wrote in message
news:1111709918.4961a481db139edb244d3e479b4c2c3f@t eranews...
[SNIP]

The "paradox" you present is indeed an interesting case. Having not
previously come across it, it caught my attention and I spent
considerable time seeking an answer.



First I examined the question of how forces transform under a Lorentz
transformation between inertial frames. I did this by examining the
behavior of an accelerating body (actually a point unit-mass body under
constant acceleration as perceived on board the body), observed both in a
rest frame (the frame in which the body is at rest prior to acceleration)
and in an inertial "moving" frame, related by the usual Lorentz
transformation equations.



I computed the usual kinematic parameter set (time, position, velocity
and acceleration) for each frame's observer. This allowed me to also
compute the momentum-per-unit- mass in each frame as well as the time
derivative of that momentum.



The latter is, by definition, the force-per-unit-mass that produces that
acceleration.



The results clearly show that although observers don't agree about
position,velocity and acceleration, observers in all inertial frames will
agree that the force (the thrust being used aboard the body to accelerate
that body) has the same constant value.



Conclusion: Force (or at least this type of mechanical force) is an
invariant under a Lorentz transformation. [This may indeed be true for
total EM force as well, though the electric and magnetic components vary
between inertial frames.]



But then, why isn't there an unbalanced torque on the bracket in the
moving frame, since clearly the bracket leg along the frame velocity is
foreshortened from its proper length?



The answer is the following, and strangely, it follows from our
understanding of relativistic stellar aberration.



To the observer in the "moving" frame the angle between the two bracket
legs is not a right angle, as it is in the "rest" frame. Think of it
this way: If the "at rest" observer were stationed on the bracket hub,
sighting his telescope along the transverse leg and saw star 'A' (a
coincidence which comprises an event in spacetime), the "moving"
observer, sighting on star 'A' (and therefore also along that leg's
length), would find his telescope to be directed at a different angle
relative to the direction of motion.



If PHI is the angle between velocity direction and star in the "rest"
frame, and "phi" in the "moving" frame, then



tan(phi/2) = sqrt{[1-(v/c)]/[1+(v/c)]}*tan(PHI/2).



[This equation results from an application of the Lorentz transformation
equations.]

In the "rest" frame, PHI = 90 deg so that phi = 2* arctan(
sqrt{[1-(v/c)]/[1+(v/c)]}).

The leg rotation is (90 - phi) deg so that the transverse torque lever
arm is F*L*cos(90-phi).

If you do the computation you'll discover that cos(90-phi) =
sqrt[1-(v/c)^2] for all v.



This means that in the "moving" frame the transverse bracket leg is
"rotated" by an angle such that the leg's projection on the "moving"
frame's y-axis is the same length as the other leg's foreshortening.



The zero torque in the "rest" frame remains a zero torque in the "moving"
frame.



Thanks for presenting this question. I learned a lot and it was much
fun.

Eli Botkin