On the superposition of forces

Question

Isn't force just a vector quantity? Don't vectors of the same kind add according to the superposition principle? So why don't all forces obey the superposition principle?

Answer 1

Don't vectors of the same kind add according to the superposition principle?

All forces are indeed vector quantities as is dictated by geometry (force equals to change of momentum and change of momentum is vector due to geometry)

Now, force being vector quantity only means, that I can measure its components in directions of some basis vectors, say $\vec{e}_x, \vec{e}_y$ and $\vec{e}_z$, to get 3 numbers $F_x$, $F_y$ and $F_z$ and the force will be $\vec{F}=F_x\vec{e}_x+F_y\vec{e}_y+F_z\vec{e}_z$. This is law of superposition for vectors. It tells me, that I can study different directions of motion independently.

But when we talk about superposition of forces, we do not mean independent directions, but independent interactions.

Imagine situation with 2 fixed bodies and some test body whose motion we study. Let us assume the interaction is given only by positions $\vec{r}_1$ and $\vec{r}_2$ of these two fixed bodies and position of test body $\vec{r}$. Then classical physics assumes that we can get the force $\vec{F}$ from these three positions by some algorithm, which defines function $\vec{F}(\vec{r}_1, \vec{r}_2, \vec{r})$.

Input quantities (positions) to this function are vectors. Output quantity (force) is also a vector. But this tells you nothing about the kind of form that the function $\vec{F}(\vec{r}_1, \vec{r}_2, \vec{r})$ has. For interaction to abide by law of superposition this function needs to have the form $\vec{F}(\vec{r}_1, \vec{r}_2, \vec{r}) = \vec{F}_1(\vec{r}_1, \vec{r})+\vec{F}_2(\vec{r}_2, \vec{r})$ which tells me, that I can study interactions with each of the two fixed bodies independently.

So to recapitulate: Force being vector means, that given the interaction, we can study different directions of motion independently. Principle of superposition means, that we can study interactions themselves independently.