Thank you all for the contributions above. I didn't bother to quote everyone, but I think the general conclusions I have drawn from this discussion are these:
Assuming that we are dealing with an isolated system and that all interactions are two-body, Newton's Third Law and the Conservation of...
I think what Jano is trying to say is that, in a truly axiomatic system, if some claim about nature is derivable from some other claim about nature, they should not both be considered laws (i.e. axioms or postulates). However, in practice they often are because the choice of which is a law and...
So then can it be said that mechanical energy conservation (when it applies) is a direct consequence of Newton's Laws, whereas total energy conservation is a more fundamental and independent postulate?
Thanks for the suggestion! I will definitely take a look at the videos when I get a little bit of time. Is there perhaps a "short answer" to my momentum and energy independence question, or does it maybe depend upon the system under consideration?
Yes, but in the traditional derivation of the...
Everything you said is right, of course. Sure it may not be "worth it" to consider energy conservation in an elastic collision, but that is not a purely mechanical process (energy is dissipated as heat).
I'm really asking this as an exercise in mathematical physics than in application. You are...
Thanks for the replies!
Well, if both are a direct consequence of Newton's Laws, then both should be applicable at least whenever Newton's Laws are. I don't see how, in the consideration of only classical "slow-moving" point particles undergoing solely mechanical interactions, one can be more...
Suppose we take the three Newton’s Laws as axioms.
Existence of inertial reference frames
F = ma
F(A on B) = -F(B on A)
Also suppose also we are considering purely classical mechanical processes on point particles (no heat transfer, etc.).
It is clear to me that the conservation of momentum...
Ozgen and DivergentSpectrum, thanks for the replies.
What exactly is meant by "unit density flow?" I cannot find a reference to this anywhere else.
The resource I was using is this: http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx
The discussion on divergence is copied almost...
I'm trying to figure out what the physical meaning of divergence is for a vector field.
My textbook offered the following example: if v = <u, v, w> represents the velocity field of a fluid flow, then div(v) evaluated at P = (x, y, z) represents the net rate of the change of mass of the fluid...
If someone asked me to differentiate x^2cos(x) (with respect to x implied because it's the only variable in the expression), I would do what you did in (2).
So, differentiation = the act or process of taking a derivative.
Now, whether "finding the differential" can be referred to as...
Chet,
Of course it hasn’t limited your ability to get correct answers - we’re only discussing definitions and not some behavior of the universe after all!
These terms seem to be very ambiguous, even in standard thermodynamic texts. Smith, Van Ness, and Abbott (which I know you are fond of)...
Chet,
I'm a little confused by your definition of quasi-static.
From the definition that Useful nucleus gave in the first post, it would seem that the "contact with a succession of thermal reservoirs" case would be a quasi-static process whereas the "contact with finite temperature difference"...
It's rigorous because it's impossible to contradict! Forget the Zeroth Law for a minute and consider just arbitrary functions.
Let f1 = f1(x1, x2, x3, x4) = 0 and f2 = f2(x1, x2, x3, x4) = 0.
Furthermore, suppose that we know whenever f1 and f2 are 0, so is some third function f3 = f3(x1, x2...
Say you made it a point to prove every theorem and concept in physics before you every apply it yourself. Ultimately, you'd find that you'd gone far from the realm of science into the realm of theoretical math and eventually into the realm of philosophy.
Imagine your burden of proof! Not only...
This last post of yours very elegantly clarified my confusion.
My mistake, as I can see now, was to calculate change in entropy for a fall + perfectly elastic collision process and then erroneously accredit gravity for this entropy change.
Thank you very much for sticking with me through this...
The way I see it, as the object is falling, its atoms have a net directionality of movement downwards. When this is abruptly brought to a halt by collision with the ground, the vibrational energy of these molecules inevitably increase, increasing internal energy of our system.
We have assumed...
My apologies. Of course, you're right. But, I'm afraid this leaves me even more confused than before.
I was using the form: Δ(K.E.) + Δ(P.E.) + ΔU = Won + Qin.
For the actual process, I reasoned Q=0 (body is thermally insulated), W=0 (no external forces), and Δ(K.E.)=0 (block is at rest...
Chet,
Thank you very much for the detailed derivation. I rewrote the entirety of both posts out by myself and the logic makes sense to me.
One small point of simplification: you’ll notice I was specifically considering a rigid body which, by definition, must be incompressible. Thus, alpha = 0...
Chet,
Thanks for the reply.
I believe the equation you are referring to is: dS = \frac{Cp}{T}dT - (\frac{dV}{dT})_P dP (imposed with the constant temp. condition)
However, this equation (and the Maxwell relationships from which it is obtained) presupposes that entropy can be expressed solely...
Another way to phrase the question that I just thought of (based on a different thread you replied to):
I know for a simple system where there is a single force-displacement conjugate pair (pressure-volume for instance), 2 properties of the system are sufficient to specify the state.
But in an...
I am trying to reconcile the result I'm obtaining for a rigid body in a gravitational field (entropy changes with height) with what I know about flowing streams from my Engineering Thermodynamics (Smith and Van Ness) textbook (entropy depends only on pressure and temperature).
Since the entropy...
Yes, that's exactly what I'm asking.
For example: http://www.learnthermo.com/T1-tutorial/ch08/lesson-B/pg04.php
Every other thermodynamics textbook I've seen has some variation of that equation (with no explicit elevation term).
Since in the open system energy balance, the same textbooks...
I have seen in a number of thermodynamics lectures that the entropy change of a system as it falls approximately isothermally from some height h to the ground is: ΔS = mgh/T
(The proof basically has you conceive of a reversible process between the same two states where some upwards force acts...
Also, some of the new-fangled experimental physical chemistry textbooks define an "infinitesimal" change in pressure to be such that the imprecision of your instrumentation is large enough to where it can't pick up such small changes/oscillations associated with the actually finite (though...
I was following along in my Thermodynamic textbook and began playing with some definitions. In the following formulation, I somehow managed to prove (obviously incorrectly) that dq = TdS for even irreversible processes. I was hoping someone could point out where in the proof I'm going wrong...
This is a purely conceptual question that I’m having trouble understanding.
From what I understand, anytime an energy transfer takes place as work, you can say that something has done work on something else. For example, if the gas in a cylinder with a piston on it pushes back the atmosphere...
Pardon? Forgive me if I'm missing something.
s depends on x, y, and z.
x and y are related; z is independent of the two.
Thus, considering the relation between x and y, we can think of s as a function of x and z only.
s then has two PARTIAL derivatives ∂s/∂x and ∂s/∂z.
If \frac{dz}{dx}=0, why would we refer to the quantity on the right as \frac{ds}{dx}?
This is STILL only a partial derivative of s since s depends on z as well.
Is there such a thing as a total "partial" derivative?
Total Derivative as I've Been Taught
From my understanding, if we have a function s = f(x, y) where the two arguments x and y are related by another function y = g(x), then there is a great deal of difference between ds/dx and ∂s/∂x.
∂s/∂x...