Re: Did Einstein Repeal Newton's Second Law?
From: Harry (harald.vanlintel_at_epfl.ch)
Date: 09/28/04
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Date: Tue, 28 Sep 2004 09:29:37 +0200
"Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com> wrote
in message news:qEX5d.258879$dW1.13375555@phobos.telenet-ops.be...
>
> "Donald Macnaughton" <donmac@matstat.com> wrote in message
news:iTW5d.1930$tT2.384584@news20.bellglobal.com...
> > I'm a statistician, not a physicist. In an essay I'm writing
> > about scientific reasoning I would like to say whether Newton's
> > second law of motion (F = ma) was "repealed" by one of Einstein's
> > two theories of relativity. However, although I've done some
> > digging, I haven't found a good discussion of this topic. So I
> > have the following questions:
> >
> > 1. Does one of Einstein's theories repeal (or modify) Newton's
> > second law? If so, how? If not, do Einstein's theories take
> > Newton's second law as a given?
>
> A partial answer:
>
> Actually Newton's second law does not give force as mass
> times acceleration,
> F = m a
> but as the time derivative of momentum,
> F = dp/dt
>
> Newton defined momentum p as mass times velocity
> p = m v
> and velocity v as
> v = ds/dt
> hence, when mass is invariant or constant, the force is given by
> F = m dv/dt
> = m a
>
> Einstein treated mass M as dependent on the so-called rest-mass m
> and the velocity v:
> M = gamma m
> where
> gamma = 1 / sqrt( 1 - v^2/c^2 )
> so he kept Newton's second law
> F = dp/dt
> and still defined defined momentum p as mass times velocity,
> p = M v
> = gamma m v.
>
> So we could say that Einstein did not change Newton's second law,
> but he used another defintion of mass.
Nicely put! What exactly was Newton's definition of mass, do you know?
Harald
> The force is now given by
> F = d(M v)/dt
> = m d(gamma v)/dt
> = m d(gamma v)/dt
> = m [ gamma dv/dt + v d(gamma)/dt ]
> = m gamma [ 1 + gamma^2 v^2 ] a
> = m gamma^3 a
> This is only valid when the acceleration has the same direction
> as the velocity. [For other direction, see article below].
>
> Nowadays this "relativistic mass" M is a bit out of fashion, and
> momentum is *directly* defined as
> p = gamma m v.
>
> You will find a very good read in this FAQ article:
>
http://hermes.physics.adelaide.edu.au/~dkoks/Faq/Relativity/SR/mass.html
>
> Hth,
> Dirk Vdm
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