A wave travels along a rope towards an end that is attached to a frictionless rod with a ring. The book tells me that the force on the end of the rope is zero because if it weren't, the end would experience infinite acceleration because it has zero mass. I don't buy this, since this argument equally applies to any infinitesimal part of the string. So the rope should never move! What am I doing wrong? (I don't doubt the force is indeed zero, I just don't buy the explanation.) #physics

@sten Assuming we're talking about a transverse wave I think there's no force on the end from the rod because it can't apply (transverse) force without friction. There is still force on the end of the rope from the rest of the rope.

If you model the rope as a system of masses and springs constrained to move transversely, the only force on the last mass is the spring interaction with the next-to-last (and you should get the linear wave equation in the limit).

@dylan My main problem is, if the vertical force on the end of the string is zero, how does it ever move upward? Either newton's first law is hereby disproved, or there is something wonky with the explanation. I think we can take Newton's first law for granted, so it's probably the latter :-)

@sten I think part of the problem is that the "end of the string" isn't really a physical object in the limit. It doesn't have length or mass and can't respond to a force.

So the "force" isn't meaningful except as a justification for the boundary condition du/dx = 0. They're using a post hoc argument that other b.c.'s would give unphysical solutions.

In the discrete approx. though, it's clear the last segment is horizontal. Probably some form of continuity argument can show this is preserved?

@dylan I agree that "post-hoc" is probably the source of the problem. But limits are being done in physics all the time, so if dx -> 0 weren't a physical thing, then we also couldn't e.g. define velocity as lim_{dt->0} dx/dt. Sigh. I think I'll have to just trust the book on this one and keep on reading. Maybe it makes sense later.

@sten I think the way I look at it is that Newton's laws get you the equations of motion, but not the constraints (in this case, boundary conditions). You can derive the wave equation for the interior of the string from Newton 2, but you have to impose two b.c.'s to determine a solution, and those come from the geometry of the system.

@sten Like: you could just as easily pin the string to a point on the rod, and you'd have a different boundary condition, but Newton 2 would still determine the differential equation that governs the string. You could compute the vertical force that would be required for this, but the significance of it is that your boundary condition is u(0) = 0 instead of u'(0) = 0.

@sten Which book are you reading, by the way?

@dylan I tried Zagoskin, "Quantum Mechanics: A Complete Introduction", John Murray Learning, 2015, but I found that incomprehensible. I also tried Feynman Vol III, but that presupposes a lot of stuff, like vector calculus. I got me the exercise book to Feynman to catch up, and I'll go back to Feynman after working through Six Ideas. (I'm doing all that b/c I want to really understand quantum crypto, where I suspect a lot is pure BS.)

@sten When it comes to QM textbooks my personal favorite is definitely "Introduction to Quantum Mechanics" by David J. Griffith. It's written in a very enjoyable way unlike most other books, although I guess it does require some linear algebra 'n stuff as well.

@rjukan Thanks, I'll check it out! I don't mind the math. In fact, the math is the easy part. The problem I have with at least some intros to QM is that it's ALL math and no physics. I mean, infinite-dimensional vector spaces, Hermitian linear forms, and all the rest of it, they're cool, but what does that have to do with electrons?