Is there any systematic approach to calculate friction?

In summary, if there are a number of objects on top of each other and the bottom one being on top of a frictionless surface, and 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, then the motion of each block is determined by assuming that all blocks move with common acceleration. If frictions are exceeded, then the problem becomes very stressful and it's very hard to do the 'guessing' thing. There is probably a "race condition" that makes the solution of this problem undetermined.
  • #1
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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.
 
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  • #2
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.
 
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  • #3
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.
 
  • #4
Simon Bridge said:
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.
 
  • #5
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?
 
  • #6
FactChecker said:
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.
 

Related to Is there any systematic approach to calculate friction?

1. What is friction and why is it important to calculate?

Friction is a force that resists the relative motion between two surfaces that are in contact with each other. It is important to calculate friction because it affects the movement and stability of objects, as well as the energy and heat generated during motion.

2. Is there a universal formula for calculating friction?

No, there is not a universal formula for calculating friction. The amount of friction depends on various factors such as the type of surfaces, the force applied, and the environment. Different methods and equations may be used to calculate friction in different scenarios.

3. What are the different types of friction and how are they calculated?

There are four main types of friction: static friction, kinetic friction, rolling friction, and fluid friction. Static friction is calculated using the equation F = μsN, where μs is the coefficient of static friction and N is the normal force. Kinetic friction is calculated using the equation F = μkN, where μk is the coefficient of kinetic friction. Rolling friction is calculated using the equation F = μrN, where μr is the coefficient of rolling friction. Fluid friction is calculated using the equation F = bv, where b is a constant and v is the velocity of the object.

4. How can friction be reduced or increased?

Friction can be reduced by using lubricants, such as oil or grease, between the surfaces in contact. It can also be reduced by using smoother materials or by polishing the surfaces. Friction can be increased by increasing the force between the surfaces or by making the surfaces rougher.

5. Can friction be completely eliminated?

No, it is not possible to completely eliminate friction. However, it can be reduced to a very small amount, such as in the case of objects moving in a vacuum. Friction plays an important role in everyday life and is necessary for many processes, such as walking, driving, and writing.

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