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How do you end up with the formula for Amontons' first law of friction? ($F_s = \mu_s \times N$) or ($F_k = \mu_k \times N$). I know how to use the formula. Now I need to know how to get to it.

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    $\begingroup$ Welcome to Physics Stack Exchange. In order for someone to be able to answer this question, that person would first need to know what the symbols in your equation mean. $\endgroup$ – DanielSank Oct 8 '15 at 2:57
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The relation is empirically observed across a wide range of materials and conditions. But it is not exact. Attempting to derive a similar equation just from lower-level physics such as solid surface molecular interactions would be horribly complex.

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