Is there any systematic approach to calculate friction?

[Prem1998]

Suppose there are a number of objects on top of each other and the bottom one being on top of a frictionless surface. The masses of all of them are given. The coefficients of friction between all surfaces of contact are given. And, the forces acting on each block are given (It's not necessary that there is some force on each object, the force can be zero on some of them and non-zero on others). For simplicity, assume all forces are horizontal and the objects are blocks on top of each other( I've asked a similar question before but it was a homework question).

Now, we have to find the motion of each block.
The first thing, of which I know, that we do in this kind of problem is to assume that all blocks move with common acceleration. We find the common acceleration and then see if the maximum possible static frictions are exceeded. If they're not then all the blocks move together.

But, if frictions are exceeded then the problem becomes very stressful and it's very hard to do the 'guessing' thing. So, is there a systematic mathematical approach to this kind of problem? It would be quite shameful if there still is no such approach given that mathematics has become so advanced.

 

[Simon Bridge]

Consider the simplest case where there is just a block on a horizontal surface with friction and under gravity - you are given the mass of the block, the static and kinetic friction coefficients, and an applied force F. You are asked to find the acceleration of the block due to F.

If $\mu_k < F/mg < \mu_s$ then there are two possible solutions...
... because, if the block is already in motion at the time force F starts to be applied, then $a= F/m - \mu g$;
but if the block is not in motion to start with, then the acceleration is zero, since $F< \mu_s mg$

In other words - to solve the problem as written, you also need to know something about the past history of the block.
Since the problem cannot be solved explicitly for the case of only one block, we cannot expect to be able to solve it for a stack of blocks.

Without that, you can still work out a plausible result by reasoning through the possible combinations of applicable pasts (like you had to do with the stacked blcok problem) to see which one seems most plausible given the physics you know. The systematic approach for this is just to go through every possible combination and crunch the numbers. It is usually easier and quicker to make informed guesses and train your intuition.

 

[FactChecker]

There is probably a "race condition" that makes the solution of this problem undetermined. Suppose you initially have all the blocks with static friction and assume that some forces are great enough to cause some of the contact surfaces to transition to dynamic friction. The solutions will often depend on which contacts transition to dynamic friction first. So there can be multiple solutions.

 

[Prem1998]

Consider the simplest case where there is just a block on a horizontal surface with friction and under gravity - you are given the mass of the block, the static and kinetic friction coefficients, and an applied force F. You are asked to find the acceleration of the block due to F.

If $\mu_k < F/mg < \mu_s$ then there are two possible solutions...
... because, if the block is already in motion at the time force F starts to be applied, then $a= F/m - \mu g$;
but if the block is not in motion to start with, then the acceleration is zero, since $F< \mu_s mg$

In other words - to solve the problem as written, you also need to know something about the past history of the block.
Since the problem cannot be solved explicitly for the case of only one block, we cannot expect to be able to solve it for a stack of blocks.

Without that, you can still work out a plausible result by reasoning through the possible combinations of applicable pasts (like you had to do with the stacked blcok problem) to see which one seems most plausible given the physics you know. The systematic approach for this is just to go through every possible combination and crunch the numbers. It is usually easier and quicker to make informed guesses and train your intuition.

The initial condition is that all the blocks are at rest initially. There is no relative motion between any of the contact surfaces initially.
I agree with you that we need some more information to solve this, maybe we can't work it out in terms of variables and we need the actual values in numbers.But I can't figure out how to solve this using intuition. Maybe I could learn from you. Will you please prepare an example for me and solve it here yourself? The example should involve at least 4 or 5 blocks. There should be horizontal forces on two or three of the blocks and no forces on other blocks. You can give random values to the variables. But the example should be such that the maximum frictional forces between two or three of the blocks exceed when we assume common acceleration of the blocks. Maybe then I could figure out what's the next step that you do from there.

 

[FactChecker]

I think that a problem like this can have multiple feasible solutions. Are you looking for a systematic method to find them all or just to find one?

 

[Prem1998]

I think that a problem like this can have multiple feasible solutions. Are you looking for a systematic method to find them all or just to find one?

Like I said earlier, now i understand that this problem can have multiple solutions. But, can you please prepare a problem here involving 4 or 5 blocks which has a unique solution. Can you please give random values to all the variables such that frictions are exceeded when common acceleration is assumed? If you solve just one such problem here, then it will be easier for me to work out what's the next step that you do by intuition after the common acceleration assumption fails.