Consider the following situation. Two blocks A and B connected by an inextensible string rest on a rough horizontal surface. The mass of A is 2kg and that of B is 3kg. The coefficient of static friction between block A and the surface is 0.1 and the coefficient of static friction between block B and the surface is 0.2
Two external forces of magnitudes 1N and 8N are now applied on A and B. The situation is shown in the figure below:
My goal is to find the tension in the string connecting the blocks and the friction acting on each block after the forces are applied.
First, I calculated the values of limiting friction corresponding to A and B, using (friction coefficient)*(normal contact force). For A it turns out to be 2N and for B 6N.
Now, considering the two blocks plus the string as a single system, the total external force is 8-1=7N rightwards and the maximum possible static friction is 6+2=8N leftwards. From this I concluded that the system cannot accelerate under the application of the given forces. Moreover, the sum of the static friction forces acting on A and B must be exactly 7N in order to balance out the external force.
To find the tension in the string, we need to find the individual magnitudes of the two friction forces. But I can't think of any reasonable way to obtain this using Newton's laws. I would like to know the right way to approach this problem.
The correct answer:
- Friction on block B is 6N leftwards.
- Friction on block A is 1N leftwards.
- Tension in the string is 2N, the system is in equilibrium.
From the solution, it seems that the friction on block B has been taken as the limiting(or maximum) value, 6N.
How did they arrive at this conclusion? Is there a way to explain this just using Newton's laws and motion constraints, or is some additional reasoning and extra conceptual insight required? Are these the ONLY possible values of friction forces? It seems like the forces could be anything as long as they add up to 7N and do not exceed their respective limiting values(although my gut feeling tells me that only one solution is possible for a practical situation such as this). The tension would then be decided accordingly, depending on the values of the friction forces (assuming that the breaking stress of the string is large enough, of course). In other words, Newton's laws predict an INFINITE number of solutions. What would be the most accurate and way of arriving at the answer to the question?
Final question: Is my way of concluding that the system does not accelerate correct? Or is there a more solid argument?