Why does the Pathagorean theorem work for forces?

[GiA]

Hey guys,

I understand why the theorem describes the relationship between the lengths of the sides of a right triangle; but how do we know that the theorem works for forces? why is the hypotenuse the resultant of those two forces( x and y).
By the way, thank you very much for having me on this forum.

 

[SteamKing]

BTW, it's the Pythagorean Theorem. It works for anything which can be described using the geometry of a triangle: Forces, distances, velocities, etc.

http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition

After all, how do we 'know' 2 + 2 = 4?

 

[Delta2]

Because in physics we make the "assumption" (rather obvious assumption i would say) that forces can be fully represented by the mathematical object of vector. For vectors and regarding the addition or substraction of vectors the pythagorean theorem holds or more generally the cosine law holds.

 

[Fredrik]

This is a much better question than one might think at first. It's not really easy to answer it.

If Alice alone is pushing with force $F_A$ on an object with mass m, its motion satisfies
$$mx''(t)=F_A(x'(t),x(t),t).$$ If Bob alone is pushing with force $F_B$ on the same object, its motion satisfies
$$mx''(t)=F_B(x'(t),x(t),t).$$ So what if they both push at the same time? Classical mechanics says that the motion will satisfy
$$mx''(t)=F_A(x'(t),x(t),t) +F_B(x'(t),x(t),t).$$ This can't be proved by mathematics alone. It's one of the assumptions we make about how mathematical concepts correspond to things in the real world, as part of the definition of classical mechanics.

Why $F_A+F_B$, rather than say $F_A+2F_B$? A partial answer is that if it would turn out that the latter formula gives us better predictions about results of experiments, then it would suggest that some sort of fundamental asymmetry is making things difficult for us. Maybe the result of the experiment depends on who's performing it. This would pretty much make it impossible to do science. Maybe there's something fundamentally different about the direction in space in which Bob is pushing, which makes his push "count" twice as much. This would contradict the principle of isotropy of space.

The formula $F_A+F_B$ is the first one that a person with some intuition about science and the properties of space would guess will be appropriate in a theory of physics. This is a good enough reason to make it part of the definition of classical mechanics. If we make such a guess with little or no input from experiment to guide us, we must of course be prepared to reject the theory we have defined, if experiments disagree with its predictions. In this case, experiments agreed, and there was no need to abandon the theory.

It may be possible to replace this assumption with a prettier one. Maybe an assumption that turns the principle of isotropy of space into a precise mathematical statement is all we need. But we can't be sure without a theorem that proves this. If there's no such theorem (I don't know if there is), then we can probably still attack each specific formula that someone suggests we use instead of $F_A+F_B$, by showing that it violates some symmetry principle.

 

[Delta2]

What you actually saying Fredrik is the justification of the assumption of why forces "directly follow" vector arithmetic (that is we can replace 2 (or more ) forces that are represented by the vectors Fa and Fb with a force that is represented by the vector Fa+Fb and still observe the same result in the physical reality). You are right this justification indeed can't be done by mathematics. However the fact that the magnitude of Fa+Fb follows from pythagorean theorem or the cosine law and that the direction of Fa+Fb is that of the third side of a proper triangle with sides Fa and Fb is proved by mathematics.

 

[UltrafastPED]

The Law of Superposition of Forces (parallelogram rule) has been known for a very long time ... far in advance of Newton. It is an empirical law, easy to observe.

There is a nice article link in this thread: https://www.physicsforums.com/showthread.php?t=751172#6

 

[Fredrik]

However the fact that the magnitude of Fa+Fb follows from pythagorean theorem or the cosine law and that the direction of Fa+Fb is that of the third side of a proper triangle with sides Fa and Fb is proved by mathematics.

Yes, the formula $|F_A+F_B|^2=|F_A|^2+|F_B|^2+2F_A\cdot F_B$ (where $\cdot$ is the dot product of vectors) is a simple mathematical theorem. When $F_A$ and $F_B$ are orthogonal, the last term is zero. This special case is the Pythagorean theorem.

 

[256bits]

Why FA+FB, rather than say FA+2FB? A partial answer is that if it would turn out that the latter formula gives us better predictions about results of experiments, then it would suggest that some sort of fundamental asymmetry is making things difficult for us. Maybe the result of the experiment depends on who's performing it. This would pretty much make it impossible to do science. Maybe there's something fundamentally different about the direction in space in which Bob is pushing, which makes his push "count" twice as much. This would contradict the principle of isotropy of space.

Ye old tug pushing a barge against/with the flow of water in a river comes to mind of an anisotropic space. Pushing in either direction will of course give the barge the same velocity, or acceleration, the case may be, in the water "ether", but certainly not with respect to the distant shoreline.

Being in a gravitational field is another situation, where if one did not know they were in such as field (ie you had your back to the planet or star ), engaging your rockets to or away from the gravitating body would give the same acceleration with respect to the "space ether" but not wrt to the distant stars.

In fact, in both situations, even by not providing a force by tugging or using rocket propulsion, one would notice a drift wrt the distance shore or distant stars.

Of course we know how to deal with such situations, by determining the flow of the river or by calculating the gravitational attraction of the planet or star, and thereby break the problem up into, what we consider to be, its separate calculable physical constituent parts.

The "ether" problem was done away with earlier in the century by Michel-Morrison ( isotropic space ) and Einstein (there is no either ).

And we are left with what we have now, which seems to work, so far so good,

 

[RyanXXVI]

In physics, math, and any other realm of logical thinking, we "know" a fact because it applies to one or both of two thought processes: inductive reasoning or deductive reasoning. Inductive reasoning is when one notices a concrete pattern that occurs every time under a certain circumstance, which is how Pythagoras found his theorem in the first place, he noticed a pattern with the lengths of a right triangle. Deductive reasoning is extrapolating previously known facts to come to a conclusion, which is like solving for speed when you know distance and time. The main difference between these two processes is that only deductive reasoning answers "why". Sadly, that involves "extrapolating previously known facts", which is why scientists have not been able to complete a theory of everything, we cannot answer all the "whys", but we can answer the "hows".

Sometime, someone just wondered if the Pythagorean theorem worked with forces, tried it, and his hypothesis was proven correct, another example of inductive reasoning. I hope that answered your question.