I showed you mine, now aren't you going to show
me yours?
Fair enough. Here goes:
Let me use a fancy object for pedagogical purposes:
two blocks connected by a spring. Also the usual table.
Step 1: Impart energy to the object while it is disconnected
from the table. In particular, set it spinning, so that
there is
-- rotational kinetic energy in the blocks, plus
-- potential energy in the spring.
Note: The energy is not imparted to the center-of-mass
motion of the object. This emphasizes the pedagogical point
that there is nothing special about the center-of-mass mode.
I can choose any mode that has lots of energy but low
entropy.
Note: I have imparted energy while the object is disconnected
from the table, so that the blocks can be treated as pointlike
(no relevant internal structure) to an adequate approximation
_during this step_ of the operation.
Step 2: Let the spinning object touch the table. It will
spin around for a while and dissipate its energy as heat.
Step 3: Verify that the rise in temperature is given by the
obvious formula, delta T = E / C where C is the total heat
capacity (object plus table) and E is the total (kinetic plus
potential) energy imparted to the object in step 1.
Note: I have not said anything about the details of the
frictional process. I have not calculated the transfer of
anything across the block/table boundary, not Q, not W.
Note: I have not proved that the increase in thermal
energy takes the form of kinetic energy... which is good,
because that isn't true. It's bad luck to prove things
that aren't true.