From crose Thu Oct 16 22:06:28 1997 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA28800; Thu, 16 Oct 97 22:05:39 EDT Date: Thu, 16 Oct 97 22:05:39 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9710170205.AA28800@MOGLI.rutgers.edu> To: 330_501 Subject: potential energy Cc: crose@MOGLI.rutgers.edu Status: RO Hi Folks, I had a LOT of fun tonite! It's fun to mix it up and get a colloquy going. I hope it's not too disturbing to you when it feels like we're on shifting sands. Just rest assured that the confusion is sown with the best of intentions and I'll always stop playing around and clear it up if it gets ugly. If you HATE that, then let me know and I'll tone it down (if enough folks want me to). Otherwise get ready for a wild ride. Now remember about potential energy? Remember when I dismissively said, ENERGY IS POSITIVE! I realize I may have come on too strong there. It was really more of an incredulous question than an answer; i.e. what do you mean that ENERGY is negative!?!?!?!? Can you have negative mass (E = mc^2) !?!?!?!!? What's going on here!?!?!?!?!? So the question to you is: Is potential energy ALWAYS NON-NEGATIVE? Certainly kinetic energy is non-negative (the zero reference point is always a mass at rest in an inertial frame). However, is this always true for potential energy where you choose the reference as the zero force point? More importantly did I give away 50 cents / head in class or did I save 50 cents/head? Let's say I was goign to wager $50 a head. Would I have changed my answer? I'm not telling :) Just trying to make it interesting .... :) If it takes money... :) The ONE THING I'M NOT GOING TO MISLEAD YOU ON is that potential energy is almost ALWAYS referenced to the zero force point. At the zero force point we have zero potenial energy. The potential energy stored is the amount of work done when an object is moved from a given point to the zero force point. Remember how work is defined... integral F dot ds a contour integral of the dot product of the force vector F and the displacement path tangent ds Now, can WORK be negative or must it always be positive????? :) :) :) I JUST LOVE THIS JOB!!!!!!!! :) :) :) :) ***************************** Now as for system analogies. Let's look at the constitutive relationships of various physical components: F = ma = m x_dot_dot F = Bv = B x_dot F = Kx I = C v_dot = phi_dot_dot I = V/R = phi_dot/R I = phi/L For the way things were written here, we have capacitance analogous to mass, resistors analogous to dash pots and inductors analogous to springs (force -> current displacement -> flux (integral voltage) ) But LOOKY HERE! We could also have V = Q/C V = Q_dot R V = L Q_dot_dot in which case we have capacitance analogous to springs, resistors ARE STILL ANALOGOUS TO DASH POTS and inductors analogous to mass (force -> voltage displacement -> charge) So going back to the example in class we had a diffeq in VOLTAGE of Rv + L v_dot + RLC v_dot_dot = blah blah And OH MY GOODNESS! The terms don't seem to match up!!!!!!! Well, let's divide by RL v/L + v_dot/R + C v_dot_dot = blah blah2 OH MY GOODNESS AGAIN! This is just the same as phi/L + phi_dot/R + C phi_dot_dot = int(blah blah2) dt after we integrate both sides. (our first option of mass=capacitance and inductance = spring and I -> force and flux (integral v) -> displacement). The bottom line is you ALWAYS map energy storage elements to energy storage elements and dissipative elements to dissipative elements (like dashpots to resistors). For thermodynamic systems... Let Q be the heat flux and T the temperature. An insulator has Q = TR where R is the insulation coeff A mass of stuff has T_dot = QC where C is the thermal capacity There are no alternate energy storage elements in thermo... if you want temperature oscillations you have to use active elements. Cheers, Chris Rose PS: By the way I'd really love to see a workup of Hamiltonians for dissipative systems (HINT HINT). I've not found one simple enough to teach as part of a lecture.