Why is Pythagorean theorem applicable to forces?

Question

This question has been bothering me for a while now. It is relatively simple question but I can't find the answer. If I have a force of 3N in the Y-axis and a force of 4N on the x-axis why can I use the Pythagorean theorem to solve it to get a force of magnitude 5N? I mean isn't Pythagorean theorem defined for lengths. Also, why can I break down a force into its component using trigonometric functions, aren't those derived from lengths as well?

The question may be stupid but could someone explain what the problem is in my understanding.

Answer 1

Forces are defined in Newton's second law, and they are implicitly defined as vectors. The reason is that the right side is a vector: $m\overrightarrow{a}$. This is a vector because it is the second derivative of a vector. the left side of the equation is the force, and for consistency it must be a vector. Ultimately, it is experiment that would decide if a theory is good at modeling data or not.

Answer 2

A force is a physical thing. It is something we feel, something we can exert blablabla. When we want to do calculations, we decide to model forces using things called vectors, which for introductory purposes, you can imagine as being a certain arrow in the plane, having some length, and pointing in some direction.

So, in your case, you're taking a physical problem, and formulating it mathematically. Once you have formulated the problem as a mathematical one (here it's a simple geometric problem), you have available to you all the familiar results of geometry/mathematics (Pythagoras, trigonometry, calculus, vector-algebra, vector-calculus and so on).

At the end of the day however, if you're asking why such a model of what we call force works, then the answer is by experiments. After all that's how science works: we wish to investigate something (in your example force), so we make a model for it (a simple definition of vector), and then check our model against the observations of the experiment, and then conclude whether or not our model is good for the intended purposes.

Answer 3

The Pythagorean theorem is a mathematical statement we just intuitively attach the unit length to it. More over in a vector space the same statement holds. It's about how we define the magnitude of a vector. A vector $\mathbf{r} = (x, y)$ has a magnitude of $r = \sqrt{x^2 + y^2}$ or alternatively the norm is $r^2 = x^2 + y^2$.

Answer 4

To answer your question, we need to first understand what a vector is: In physics, a vector is defined as any quantity with a magnitude and a direction. So, a force is a vector, well, because it is defined as a quantity with both a magnitude and direction. It is easy to visualise this: As you have mentioned in your post, there can be a force of 3N in the Y-axis. In this case, the magnitude is 3N and the direction is in the direction of the Y-axis.

Ok, so now that you understand what is a vector, the question is why can you break a force down into its x-y component and why can we use Pythagorean theorem to sum up the components? Basically, a simple fact about vectors is that any 2 perpendicular vectors does not affect each other. By that, I mean that no matter how the magnitude of one vector increases, the magnitude of the other vector won't be affected. You can visualise this with a simple thought experiment: Imagine you have a box and you apply a force vertically upwards, will this box move along both axis? (The answer is no) Conversely, if you apply a force diagonally, the box will move along both axis. The purpose of this experiment is to convince you that 2 perpendicular vectors won't affect each other.

After we have derive the above conclusion, people generally decide to reslove vectors into its x-y component, because this 2 components will not affect each other, thus making calculations easier. You can indeed resolve vectors about any other axis that may not be perpendicular to each other, but that will just make your life tough (try doing it if you don't believe me).

As for the question on isn't Pythagorean theorem and trigo for length, well, vectors have length(magnitude) component in them, so they can be applied. Unfortunately, Pythagorean theorem only tells you the magnitude of the force vector, you have to find its direction using vector addition.

Answer 5

It is not stupid question at all.

As @Wolphramjonny said, in Newton law's we say that force is equal acceleration times mass. Mass is assumed to be scalar, i.e. just a number that characterizes how much resistance body has against an acceleration. Acceleration is vector, and it needs to be one due to geometric considerations.

If we define force as $$\vec{F}=m\vec{a},$$ it indeed needs to be a vector. In this sense it is trivial fact that has no deeper, physical meaning.

But force is not actually defined by the law $\vec{F}=m\vec{a}.$ Force is defined by other means, it characterizes interactions between bodies. The law $\vec{F}=m\vec{a}$ tells us how to use the force if we have it or helps us to find it if we don't. But the true law that defines force comes from interactions. One example is gravitational law:

$$\vec{F}=-G\frac{mM}{r^3}\vec{r}$$

The implicit claim is, that interactions between bodies is well described by vectorial quantities. This is pretty nontrivial statement. It tells us for example, that if we have two massive bodies that interact gravitationally with the third, the resulting force is the vectorial sum of both forces. Or in other words the two forces are independent and are not influencing each other and they have equal "vote" in how the body they act on will move.

This is not true in general. In stronger gravitational fields, the forces become coupled and we can no longer compute force on a body by simply adding the constituent parts.

We have two choices to tackle this. One is to keep insisting that force is supposed to be vectorial quantity. Then the force law would become complicated expression. We would no longer be able to write gravitational law as simple interaction between two bodies, but we would need to include the coupling between the bodies. Or we can stop claiming force as a vectorial quantity is a good way to describe gravitation and we start seeking another theory. This is the path Einstein took, when he started seeking General Relativity. But since force and its equation $\vec{F}=m\vec{a}$ was (and still is) a useful concept in many scenarios, he did not confuse his peers and students by redefining the word force, instead he simply stopped using it.

The similar thing happened in electromagnetism. We stopped using force as a good description of electromagnetic interactions, because there the force is coupled with the past (it takes time for the information to reach your charged particle) and instead we started using fields for describing the nature.

We can always write force as a vectorial quantity, but this might not always be the best idea if the interaction is strongly coupled. But if the coupling is weak, then it is a good idea. The fact, that force has such a prominence in mechanics is due to the fact, that most researches back in the day were doing research with gravity and forces that are indeed weakly coupled. In fact, it was the explanation of planetary movements that triggered scientific revolution. And gravity in our solar system is weakly coupled, the resulting effect of gravitational interaction on planets and asteroids is indeed a simple sum of effects of individual planets (or Sun).

PS

I am using the word "force" in two different meanings. Sometimes I mean the force as quantity that obeys $\vec{F}=m\vec{a}$ and sometimes I use it more generally as quantity that describes interaction between two bodies. I do not have time right now, but if it is confusing, let me know and I will try to reformulate my answer later.