-1

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?

Qmechanic
  • 201,751
Ahmed Samir
  • 458
  • 3
    Forces do obey the superposition principle. The net force on an object is the vector sum of the individual forces acting on it. – interoception Apr 30 '22 at 18:25
  • Yeh like interoception said, all forces obey the superpostion principle not sure where you have seen otherwise but this is a fundamental rule to remember as it tends to leads to other rules such as the superposition principle of electric fields etc. – Dirac Delta Yeah Apr 30 '22 at 18:39
  • yes but why do physics books always say that it shouldn't be obvious and that it's just an experimental observation? – Ahmed Samir Apr 30 '22 at 18:55
  • An in depth treatment of that question was carried out here. Which books say that it shouldn't be obvious and that it is just an experimental observation ? – Kurt G. Apr 30 '22 at 18:58
  • 1
    "The principle of superposition may seem “obvious” to you, but it did not have to be so simple: if the electromagnetic force were proportional to the square of the total source charge, for instance, the principle of superposition would not hold, since (q1 + q2)2 - = q2 1 + q2 2 (there would be “cross terms” to consider). Superposition is not a logical necessity, but an experimental fact" – Ahmed Samir Apr 30 '22 at 19:08
  • from griffith's electrodynamics – Ahmed Samir Apr 30 '22 at 19:08
  • Thanks . In my youth I started to think as follows : the motion of a particle can be thought of being a series of infinitesimal translations that can be described by vectors that allow superposition. Taking a time derivative we find that velocities allow superposition. Taking another time derivative we find that acceleration allows superposition. By Newton's second law we find that the force in that case allows superposition. This is of course a very limited argument that at best holds only in a special case. Note however that the EM force can be measured as force on a test particle. – Kurt G. Apr 30 '22 at 19:32

1 Answers1

2

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.

Umaxo
  • 5,818