symplectomorphic
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 Jan 17 comment Block on a block problem, with friction @Chris: would it be wrong to say that in this problem we can't DERIVE the fact that for a sufficiently small force the accelerations will be equal, and for a sufficiently large force won't? Jan 17 comment Block on a block problem, with friction @Leo: no, the top mass doesn't have to be less than the bottom mass (at least I don't see why you would think it has to be). And I'm not talking about a time-varying force. I'm just talking about the dynamics as a function of different applied forces. For any two-block system, I claim there is a sufficiently large force you can apply to the top block to make the top block accelerate with respect to the bottom block, no matter their masses. Jan 17 comment Block on a block problem, with friction @Leo: Imagine a small book on a hockey puck on an air hockey table. If you apply a small force to the book, the whole system moves, but clearly there is a force you can apply to the book to make it slide off past the puck. I guess the point in this problem is you first have to make dynamical assumptions about the behavior of the two accelerations as a function of the applied force, and then work out all the friction forces from there. The graphs of $a_1(F)$ and $a_2(F)$ will agree up to a point $F_{max}$, with static friction holding the system together, but for $F>F_{max}$ the graphs diverge. Jan 17 comment Block on a block problem, with friction @Chris Ah, okay, so the moral is it's wrong to say that static friction always opposes an applied force with equal magnitude, and to look at FBDs in isolation when dealing with an interacting system. The acceleration assumptions have to come first, and the derivations of friction forces later. I think this is what trips me and many students up on this problem: in single block problems, you can compute the friction forces BEFORE knowing anything about accelerations. Not so here, though it's tempting to do so if you look at the FBD for only the top block. Jan 17 comment Block on a block problem, with friction @Leonardo: no, just apply a force of magnitude greater than what I derived for $F_{max}$. Then the top block will move with respect to the bottom block and kinetic friction takes over. Jan 17 comment Block on a block problem, with friction I'm still confused, because of the opposite case I mentioned in my comment above. Is my intuition wrong that the two blocks should still accelerate together when $F$ is less than the max static force? If not, then I don't understand how the accelerations in the equations could both be with respect to the ground. That's why I ask. Jan 17 comment Block on a block problem, with friction Can you elaborate a bit, or spell this out exactly? I don't know what it means to say "when you look in that reference frame, you're still below the max static friction." I'm thinking about free-body diagrams. Consider the opposite case, where a force $F$ of magnitude less than the max static friction force is applied. Then the sum of horizontal forces on the top block gives zero acceleration, but the reaction friction force on the bottom block gives a positive acceleration -- & yet shouldn't the blocks be accelerating together in that case? I want a way to be totally rigorous about this. Sep 26 comment Am I making the right assumption about a jump discontinuity in the acceleration? Found my algebra mistake after reading through my work for about the fiftieth time. Thanks for confirming I wasn't losing my mind about the general approach. Sep 26 comment Am I making the right assumption about a jump discontinuity in the acceleration? brachistochrone* Sep 26 comment Am I making the right assumption about a jump discontinuity in the acceleration? I think your change of title is misleading -- it's a much less interesting question. The question is a brachiostone-style time minimization problem: the point of putting "time minimization" in the title was to bring out this connection.