work and energy

Prompted by John M, I have re-read his paper as well as the Bernard & Sherwood paper. In fact I did read both the last time this topic came around on PHYS-L, but things don't really stick until you really have to use them. Like in teaching plainly, if that is possible.

I have written up a long summary of the whole topic. So far I have worked (excuse the pun) through levels one and two below, but I'll include my thoughts on level three to see what you think. Before I spring a URL for my whole article, let me start by summarizing.

Level _one_ = the whole system is one object of mass M and (center-of-mass = cm) speed V. I don't concern myself with any internals. We easily derive: W_cm = delta K. Here W_cm is the cm work or pseudowork (ie. integral of F_net dot cm displacement) and K_tr is the cm translational KE (ie. MV^2/2). This view is only useful for a few problems, but is a helpful conceptual starting point because we can derive this equation exactly from Newton's laws and kinematics.

Level two = I break the universe into macroscopic parts (blocks, strings, pulleys, earth, etc) but I ignore their internals. Now I can talk about forces between parts. If conservative, I associate this with PE. For simplicity, let's do this even if one of the two interacting parts is outside the system. (If this makes your skin crawl, just choose your system to include every interacting part.) Now sum over parts, sum delta(K_tr_i) = sum(W_cm,nc_i) + sum(W_cm,c_i). Move the last term to the left and redefine W_nc = sum(W_cm,nc_i), delta K = sum delta(K_tr_i) [K is NOT equal to K_tr] and delta U = -sum(W_cm,c_i). This gives W_nc = delta E_mech, with E_mech = K + U. Call this the work-energy theorem. This is standard textbook stuff and works just fine for textbook problems (even when nasties like friction are present) provided one essential point is emphasized:

Thermodynamic work is hereby defined as the integral of a force and the displacement of its point of application. Define W_thermo for an object as the sum of these integrals over all forces acting on the object. Note that we cannot relate this to the net force nor to the displacement of the cm in general, although in some useful special cases we can.

Level three = we break the system down to its ultimate microscopic particles. (Whether this means small chunks, atoms, or nuclear particles is determined by the requirement that all relevant energy exchanges have been accounted for, ie. there's no need to include nuclear particles if the nuclear degrees of freedom are not being accessed.) I am going to boldly make three claims at this level:

(i) There is no heat transfer. All energy exchanges are ultimately mediated by forces between particles. I am here trying to answer to John D's and Jim G's desires to avoid any Q term. (Maybe I'll have more to say after tracking down the Barrow article.) So I'll write the first law of thermo as W_thermo = delta E_mech + delta E_int. Here I've arbitrarily carved out some of the energy and called it mechanical. This includes K and U of the level two parts, and at the risk of further confusion, nothing stops me from re-defining K *again* and including the rotational energies of the parts too. (I can do this at level two also, by calculating the works done by the torques.) After all, this is what we usually do in mechanics once we hit the rotations chapter. (Vibrational kinetic energy is a bit troublesome because I don't know whether a spring is a part, or a sum of parts. Let me ignore that issue for now and return to it in a later post.)

(ii) For every force at the particle level, W_cm = W_thermo, because by definition my particles have no relevant internal structure. I *here* make contact between the work-energy theorem and the first law of thermo. Note that when treating say Atwood's machine, the particles are the blocks, but when treating sliding friction of a block, the particles are the asperities, surface layers, and the remaining bulk (or something vaguely like that - subdivide further if you think that's necessary, I won't quibble over details).

If you just want to do mechanics and never discuss heat, internal energy, etc you can stop at level two. You will be able to solve all problems just fine using the work-energy theorem. Don't listen to any naysayers who claim you can't do work by friction, skaters pushing off walls, etc. You can do them all fine - just remember to calculate the cm work not the thermo work!!!

However, if you want to go on to discuss questions like what happened to the mechanical energy dissipated by friction, then you need to introduce the first law. You will not be able to do this in any problem where it matters (egs. sliding friction, ball of clay falling to the floor, adiabatic compression of an ideal gas, etc) unless you are cognizant of *some* microscopic issues. This doesn't mean on first encounter that you have to become an expert on quantum mechanics or something! But you are going to need to know a few things about internal energy of an ideal gas, asperities on the bottom of a block, etc to be able to go beyond merely stating the obvious and rather trite fact that "all the lost mechanical energy ends up as internal energy."

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work and energy - take 2