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Leigh's paradox (Re: work done by friction)



Leigh tells us that only two situations are possible, either
VAf=VBf=0 ( brick B is sufficiently long to stop A) or
bricks separate before their initial kinetic energies are
thermalized. Consider each case separately.

1) Final velocities zero. Magnitudes fAB=fBA=f=const.
-------------------------------------------------------------------
In this case the only unknowns, in his equation (3), are TA
and TB. This is the energy equation and I do not want to
use it. The second equation becomes 0+0=0 and the first,
as before, simply defines the coordinate system of reference.
Knowing the accelerations of A and B (force/mass) we can
calculate the sliding distance d and get the work done by
each force. Knowing d we can find TA and TB from

CA*(TA-Ti)=f*d and CB*(TB-Ti)=f*d

I would expect the values of TA and TB to satisfy the energy
equation which I did not use. I am assuming that Ti, CA and
CB are given and that their temperature dependence is
negligible. Also that changes in thermal energies are initially
equal on each side of the boundary.

2) Final velocities are not zero. Magnitudes fAB=fBA=f=const.
------------------------------------------------------------------------

This is indeed a problem whose solution is not possible, unless
the values of VAf and VBf are given. To make the problem more
difficult one can specify lengths of the bricks LA and LB; in this
case one would have to calculate VAf and VBf from the given
LA, LB and constant accelerations (f/MA and f/MB). The rest
would be essentially the same as before, the well defined values
of f*d, TA and TB. Again, I would expect the energy equation
(with calculated VAf and VBf) to be consistent with the found
values of TA and TB.

Where am I wrong? What am I missing? I did not follow all the
nuances of recent debates. I am using the force and work
approach because energies are "in the next chapter", so to speak.
Ludwik Kowalski