# $μ = \tan\theta$ - Why? It makes sense mathematically, but if it's based on the materials how does that work?

We were presented with a problem in physics today in which we calculated that the coefficient of static friction is equal to the tangent of an angle

This is on a ramp using the laws of friction.

My question here is... how? Like I get that it makes sense mathematically and all, but isn't it something to with the materials? My professor gave me a cryptic answer regarding "imperfect models" but told me to figure it out. This is me figuring it out I'll post a response if I end up making sense of it.

• The coefficient of friction $\mu$ does depend on the material interface - but the force of friction $F_F$ also depends on the angle: $F_F = \mu \cos(\alpha) F$, because only the force normal to the plane is relevant. Can you please show how you derived $\mu = \tan \alpha$ in class? Oct 8, 2018 at 15:19
• Yeah, sure - This is one of those problems where a ramp raised to the point right before the box on top begins sliding. Oct 9, 2018 at 14:53
• engineerstudent.co.uk/Images/free_body.svg.png Oct 9, 2018 at 14:56
• engineerstudent.co.uk/Images/free_body.svg.png - that's the diagram of the situation, we took the formula $F_f = μ_s \times F_n$ and applied the definition of a force to calculate that relationship as follows. \begin{align*} F_f &= \mu _s \times F_n\\ (Sin \text{ } \theta \times ma) &= \mu _s \times (Cos \text{ }\theta \times ma)\\ \\ \frac{Sin \text{ } \theta \times ma}{Cos \text{ }\theta \times ma} &= \mu _s\\ \\ tan\text{ } \theta &= \mu _s \end{align*} Oct 9, 2018 at 15:08
• Ah, I see. In that case Luke's answer explains it nicely. Oct 9, 2018 at 16:06

You should read the equation $$\tan \theta = \mu$$ like this: "There is a ramp angle that will cause an object on the ramp to start to slide. The size of this angle depends on the coefficient of friction of the ramp and object. The dependency is given by $$\tan\theta = \mu$$."
It may help clarify the equation to give the angle a name, like $$\theta_{\text{slip}}$$, so that you can remember the specific meaning of that particular angle.