Retteb

2011-01-19 15:34:48 UTC

"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 in

http://einsteinhoax.com/re511.htm. 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 also 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 in

http://einsteinhoax.com/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.

All of the Newsposts made by this site may be viewed at the

http://einsteinhoax.com/postinglog.htm.

Please make any response via E-mail as Newsgroups are not monitored on

a regular basis. Objective responses will be treated with the same courtesy

as they are presented. To prevent the wastage of time on both of our parts,

please do not raise objections that are not related to material that you

have read at the Website. This posting is merely a summary.

E-mail:- ***@verizon.net. If you wish a reply, be sure that

your mail reception is not blocked.

The material at the Website has been posted continuously for over 8

years. In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE

MATERIAL PRESENTED. There have only been hand waving arguments by

individuals who have mindlessly accepted the prevailing wisdom without

questioning it. If anyone provides a significant rebuttal that cannot be

objectively answered, the material at the Website will be withdrawn.

Challenges to date have revealed only the responder's inadequacy with one

exception for which a correction was provided.

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

http://einsteinhoax.com/re511.htm. 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 also 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 in

http://einsteinhoax.com/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.

All of the Newsposts made by this site may be viewed at the

http://einsteinhoax.com/postinglog.htm.

Please make any response via E-mail as Newsgroups are not monitored on

a regular basis. Objective responses will be treated with the same courtesy

as they are presented. To prevent the wastage of time on both of our parts,

please do not raise objections that are not related to material that you

have read at the Website. This posting is merely a summary.

E-mail:- ***@verizon.net. If you wish a reply, be sure that

your mail reception is not blocked.

The material at the Website has been posted continuously for over 8

years. In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE

MATERIAL PRESENTED. There have only been hand waving arguments by

individuals who have mindlessly accepted the prevailing wisdom without

questioning it. If anyone provides a significant rebuttal that cannot be

objectively answered, the material at the Website will be withdrawn.

Challenges to date have revealed only the responder's inadequacy with one

exception for which a correction was provided.