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I have a block kept on a rough surface with a string attaching it to the wall. I pull the block opposite to the string with a certain force $F \leq \mu mg$ where $\mu$ is the coefficient of friction and $m$ is mass of the block. Will a tension develop in the string to resist the motion? Will friction resist the motion? Or will both contribute some amount of force for this purpose?

What happens in this situation: if I increase the force beyond $\mu mg$ beyond which friction cannot possibly resist my force?

Consider another situation, in which instead of pulling opposite to the string I pull the block at an angle to the string (still in the horizontal plane). How will the magnitude and direction of friction and tension vary as I change the force?

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Answer to the first question: This depends to some extent on the 'models' used for the forces of friction and tension. A typical model for string tension is as a restoring force that obeys Hooke's law: $$T = - kx$$ at least for a small positive extension $x$ in the length of the string, or equivalently, displacement of the block along the length of the string. You can go to the limit of the ideal string by making $k$ very large.

Similarly, friction is modeled as a force that opposes relative velocity between two surfaces in contact. You can consider it to instantaneously play a role when there is a "risk" of developing a small velocity.

Now, we must argue that, under a force $F$, the velocity developed on a free object is a stronger effect than displacement, if we were to conclude that "friction acts first". For this, let us look at what happens in a small time $dt$. The object develops a velocity $$dv = \frac{F}{m} dt$$ However, the displacement within the same time cannot be larger than $(dv)dt$, which goes as $(dt)^2 << dt$. Therefore, the displacement is really negligible in this time scale.

Thus, if we now turn 'on' both the tension and the friction, we expect the string tension to be negligible compared to the friction. The friction itself balances the applied force (if it is less than the limiting friction), and we effectively have a situation of zero tension.

Granted that this is a hand-wavy argument, but it may provide some intuition for why we take friction to act first in such problems, and why, as a real world example, you won't feel someone weakly pushing a heavy rock (on rough ground) that you may be leaning against from the other side.

For the other two questions, note that there is no limit on the tension that can build up in an ideal and in-extensible string, and this tension always acts along the length of the string.

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When the force you apply is less than $\mu mg$, the tension is not affected and does not affect the motion of the block. But if you apply a force larger than $\mu mg$, then the tension in the string increases.

Obviously, the block will start its motion when the string breaks (if it is a real string).

Finally, the friction direction is always perpendicular to the plane and its magnitude depends on the angle you apply the force. In other words, depends on the component of the force perpendicular to the plane. Usually, $F\cos(a)$, where $F$ is the force you apply and a the angle.

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See: The string will not develop any tension until force is applied beyond limiting friction, that is, both string and the frictional force b/w the surface will start opposing the motion only when the friction is kinetic. Hence, Static friction acts first, then comes both kinetic frictional force and the force opposing string's elongation. Thanks

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The key idea here is that Static friction is self adjusting.... It will oppose and nullify effect of your applied force F , provided maximum friction value is not exceeded.

If you imagine or draw a Free body diagram of forces on the body , you'll see that friction due to ground on Body acts in opposite direction to F.

So, Tension in string is not developed first... In fact it is only developed as a necessity when body starts sliding under your force F , ie when F> μmg.

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There may tension come into the picture first if the block shear. otherwise friction will come into the picture before tension because there must be some displacement of the block for tension to act. if block get sheared then even without relative displacement between block and surface tension will act.

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