How Do Tension and Friction Forces Interact in a Two-Block System?

[vijayram]

Homework Statement

2 Blocks connected by a string is placed on a rough horizontal floor,the coefficient of friction for block 1 is 0.2 while for block 2,it is 0.1. A force of 8N is applied on block 1 and a force of 1N is applied on block 2.Find the tension in the string and the frictional forces.
See figure.

Homework Equations

Newton's law
friction normal equation[/B]

The Attempt at a Solution

I have the solution and it says there will be limiting friction case in block 1 since the net force acts in it's direction and there is a tendency of motion towards 8N force ,but I don't understand this if there is a tendency of movement then won't both frictional force and tension act simultaneously,so how can we say friction force will be maximum.

 

[drvrm]

there is a tendency of motion towards 8N force ,but I don't understand this

if you think otherwise i.e. there seems to be tendency to move towards left side -what is the argument?

draw free body diagram and try to work out the answer/or arguments -also remember that frictional forces are 'self adjusting' and acts on 'tendencies' of motion..

 

[haruspex]

if there is a tendency of movement then won't both frictional force and tension act simultaneously

Quite so. The question is dumb, since there is no way to answer it without more information about the set-up.

In the real world, all strings have a tiny bit of elasticity, and even static friction allows a minute bit of elastic movement before yielding. To calculate what will happen, you would need to know all parameters that describe that. Even the length of string in the middle becomes significant.

 

[vijayram]

if you think otherwise i.e. there seems to be tendency to move towards left side -what is the argument?

draw free body diagram and try to work out the answer/or arguments -also remember that frictional forces are 'self adjusting' and acts on 'tendencies' of motion..

Sir,I think the argument given in the solution is not the correct explanation.But answer may be correct.

 

[vijayram]

Quite so. The question is dumb, since there is no way to answer it without more information about the set-up.

In the real world, all strings have a tiny bit of elasticity, and even static friction allows a minute bit of elastic movement before yielding. To calculate what will happen, you would need to know all parameters that describe that. Even the length of string in the middle becomes significant.

Sir,thank you for replying,if the string is assumed to be an ideal string with zero elasticity, is this explanation correct?

 

[haruspex]

Sir,thank you for replying,if the string is assumed to be an ideal string with zero elasticity, is this explanation correct?

The trouble with ideal behaviours is that they can lead to indeterminacy, as here. Consider an ideal table, four legs, on a level floor. What is the compression force in each leg? No way to tell.
The only way the real world figures out what will happen In such cases is because nothing is ideal.

 

[vijayram]

The trouble with ideal behaviours is that they can lead to indeterminacy, as here. Consider an ideal table, four legs, on a level floor. What is the compression force in each leg? No way to tell.
The only way the real world figures out what will happen In such cases is because nothing is ideal.

Yes but shouldn't we consider one if question demands it,this is a problem from a competitive examination and we always assume ideal cases.So if we assume ideal cases,will that explanation be correct?

 

[drvrm]

Sir,I think the argument given in the solution is not the correct explanation.But answer may be correct.

i think the 'argument' you are referring to is about the 'condition of limiting friction' invoked...
in your opinion what should be the frictional force on block 1 ? give Fbd also...

 

[haruspex]

Yes but shouldn't we consider one if question demands it,this is a problem from a competitive examination and we always assume ideal cases.So if we assume ideal cases,will that explanation be correct?

No. The question is a nonsense.
In an examination, with no opportunity to say 'indeterminate', the best you can do is to guess what the examiner might be thinking (and complain about the question later).

 

[mfb]

I agree with haruspex. There is a range of possible values of tension in the first example. You can calculate this range, but you cannot determine the tension.

As an easier system to study this, consider the blocks without external forces. Clearly you can have tension in the string - but the string does not have to have tension.

 

[vijayram]

I agree with haruspex. There is a range of possible values of tension in the first example. You can calculate this range, but you cannot determine the tension.

As an easier system to study this, consider the blocks without external forces. Clearly you can have tension in the string - but the string does not have to have tension.

so this question does not have determinate answer right?That is what I was thinking,friction need not be limiting.

 

[vijayram]

No. The question is a nonsense.
In an examination, with no opportunity to say 'indeterminate', the best you can do is to guess what the examiner might be thinking (and complain about the question later).

Thank you very much,I thought the same.

 

[CWatters]

I agree with haruspex. There is a range of possible values of tension in the first example. You can calculate this range, but you cannot determine the tension.

As an easier system to study this, consider the blocks without external forces. Clearly you can have tension in the string - but the string does not have to have tension.

If there is no tension in the string what stops the 8N force on the right hand block overcoming static friction (max 6N) and increasing the tension to 2N ?

Edit: OK I think I misunderstood you. Your comment that the string does not have to have tension only applies when there are no external forces.

 

[CWatters]

Anyone else agree that the range of possible tensions is 2 to 3N?

 

[mfb]

Anyone else agree that the range of possible tensions is 2 to 3N?

I got the same result.

 

[Vanadium 50]

I disagree with everyone. There is an answer with a specific tension, provided you use only the information provided in the problem. If you use other information, in particular the title of the section, "static friction", you will be confused.

 

[mfb]

provided you use only the information provided in the problem

Like the information that the coefficients of friction are static friction, as implied by "Equilibrium under static friction" in the problem statement?

There is a specific answer if you assume that the coefficients are for dynamic friction, but I don't see how this assumption is justified. This interpretation also disagrees with the given answer.

 

[Vanadium 50]

I'm wrong - there is (barely) enough friction to stop the system, so it's static. That means there is a range of allowed tensions.

 

[haruspex]

Anyone else agree that the range of possible tensions is 2 to 3N?

Yes.
One way to think about it is to allow some very small amounts of flexibility: some shear modulus in each block and some elasticity in the string.
If either the string or the smaller block is the most flexible by some ratio, the larger block will start to slip before the tension even reaches 2N. If we suppose the kinetic friction is the same, or wind back the flexibility enough to prevent a result where one block slips first, it gets to 2N.
If the larger block is much the most flexible then the tension will reach 3N and the smaller block will slip first.