Re: [Phys-l] Invariant mass and relativist mass...

["Ken Caviness" <caviness@southern.edu>]

To summarize, 

When force and velocity are perpendicular, F = gamma m a.
When force and velocity are parallel,      F = gamma^3 m a.  

The product (gamma^3 m) was at one time referred to as the "longitudinal
mass", as distinguished from (gamma m), the "transverse mass".  To me
the non-existence of one simple "relativistic mass" covering all cases
is a good reason to join (or stay in) the "anti" camp.  As has been
noted by others, these extra factors can equally well be treated as
belonging to the "a" as to the "m" in "F=ma", gamma can be seen as the
conversion to 3-vector notation, using time rather than proper time, and
more fundamentally, F=ma isn't generally true.  We consider varying mass
problems even in advanced topics of general physics:  rockets (mass
decreases), raindrops and rope/chain falling off a table (mass
increases).  I get along very well telling my advanced students that
F=ma is an approximation for F = dp/dt, and p=mv is an approximation for
p = gamma m v.  They _like_ the idea that they're getting, if not "the
rest of the story", at least "more of the story".

Ken Caviness
Physics @ Southern Adventist University

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of Tom Sandin
> Sent: Wednesday, February 27, 2008 11:41 AM
> To: Forum for Physics Educators
> Cc: LaMontagne, Bob
> Subject: Re: [Phys-l] Invariant mass and relativist mass...
>
> Look at my
> Subject: [Phys-l] Non-existence of Transverse and Longitudinal
> Relativistic
>   Mass
> post.
>
> If the students are at the |F| = m|a| level, they need to be told
> that version of Newton's second law is true only for constant mass.
>
> When the force and velocity are perpendicular, the speed and
> therefore the relativistic mass are constant--and F/a equals the
> relativistic mass.
>
> When the force and velocity are parallel, the speed and therefore the
> relativistic mass are not constant and F is not equal to ma.
>
> (F, p, v, and a are vectors in this paragraph.) For a particle in an
> inertial reference frame, the net force equals the time rate of
> change of its linear momentum, F = dp/dt = d(mv)/dt. Only if m is
> constant does this give F = m dv/dt = ma. If m is not constant, you
> have a second (dm/dt)v term.
>
> John Denker's reply is from the perspective of an relativistic mass
> opponent.
>
> Tom Sandin
>
> At 7:06 PM -0500 2/26/08, LaMontagne, Bob wrote:
> > I am still having a problem understanding why the gamma factor would
> > be applied to the mass at the introductory level. If students are at
> > the F=ma level, then the "m" that gives the reistance to
> > acceleration is really m0/(1-v^2/c^2)^(3/2), as I had noted in
> > another posting. The gamma factor is actually not the factor that
> > gives the increased resistance to acceleration - so why use it? If
> > the studets are at the Impulse=momentum_change level, then the use
> > of the gamma factor as part of the momentum seems quite natural when
> > exploring the dynamics of Relativity.
> >
> > Bob at PC
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