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Given a maximum applied force $f_{\text{max}}$ on a particle with normal force $N$, if the force is parallel to the surface then I'm told that the static friction of surface is given by $$\mu_s=f_{\text{max}}/N.$$ But I don't understand why. Is this something I can derive or is it discovered only by experiment?

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Is this something I can derive or is it discovered only by experiment?

Although friction forces can be explained on a molecular level via bonds that form between molecules, that would be too complex for most practical problems. The well-known equation

$$|\vec{f}_f| = \mu |\vec{n}|$$

is an approximation that works well enough. Here $\vec{n}$ is the normal force exerted by the surface which always acts in the direction perpendicular to the surface, $\vec{f}_f$ is the friction force that always acts tangential to the surface in the direction that opposes motion, and $\mu$ is coefficient of friction which is determined experimentally for different combinations of contact surfaces.

Please note that there are two types of friction forces, namely static and kinetic friction. The static friction acts on the body when there is no relative movement between the body and the surface and is defined via maximum value, and kinetic friction acts when there is relative movement

$$|\vec{f}_{f,s,\text{max}}| = \mu_s |\vec{n}| \qquad \text{and} \qquad |\vec{f}_{f,k}| = \mu_k |\vec{n}|$$

Coefficients of friction $\mu_s$ and $\mu_k$ are not equal, and in general $\mu_k < \mu_s$ which means that kinetic friction is less than maximum static friction force magnitude.

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Just to be clear, $f_{max}=\mu_{s}N$ is the maximum possible static friction force. If the applied force equals the maximum possible static friction force, relative motion between the surfaces is impending. For applied forces less than the maximum possible static friction, the static friction force equals (matched) the applied force. The Hyperphysics web site describes it as follows: Static frictional forces from the interlocking of the irregularities of two surfaces will increase to prevent any relative motion up until some limit where motion occurs.

But I don't understand why. Is this something I can derive or is it discovered only by experiment?

It is not something that can be derived from basic principles. It is determined by experiment. For a thorough discussion of both static and kinetic friction, I suggest you check out this link: http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html#kin

Hope this helps.

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You are already familiar with the limiting expression for the maximum force of static friction, perhaps without realizing it. If you see a book sliding across an inclined table top and you want to stop it in a hurry, what would you do? You would slap your hand on the book and push down. Why does that work?

Initially, the component of the weight along the incline is larger than $f_{s}^{max}$. By pushing down on the book, the normal force is instantly increased which in turn increases $f_{s}^{max}$ to a value higher than the component of the weight parallel to the incline which stops from the book from accelerating.

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