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A block lying on a rough surface, is connected to a wall by a mass less, inextensible string and an unknown amount of force is applied to the block opposite to the side of the wall. Now, if it is given that there is some tension in the string and the block is stationary, does it necessarily imply that limiting static friction is acting? Apparently it does. My question is, if: $$F = f_{limit} + T \rightarrow F = (f_{limit} - c) + (T+c) \rightarrow F = f_n + T_n$$

So why is it not possible that the tension in the string increases to value greater than that when limiting friction acts and the frictional force acting is lesser than the limiting friction ?

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If I'm understanding your question properly, the block in that situation is over-constrained (i.e. statically indeterminate - you cannot calculate the unknown forces from the laws of statics alone) - you cannot assume that F equals the limiting friction - both it and the tension are unknown.

When you apply a sideways force to the block, it starts to move, and as it moves both the tension (due to stretching of the string) and the friction (due to the stretching of bonds between the two surfaces, and other effects) increase. The block will move and eventually reach an equilibrium in which it is static and the sum of those two forces has a magnitude equal to the force you've applied. You need to know the two "stiffnesses" in order to determine the values of each force.

As you increase the applied force, then, at a level determined by the two "stiffnesses", the friction force will reach the limiting friction and will increase no further.

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Because the tension force of a string is a reaction force (like the friction force). This means there is no tension force until the string isn’t stretched. The string will be non-stretched until the force F is lesser than the limiting static friction force. There is a same state for friction force. The friction force is lesser than the limiting static friction force until the force F is lesser than the limiting static friction force (for example, when F is zero, the friction force doesn’t exist.)

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