F=m*a; F=K*q1*q2/r^2 ....

This question have been bugging me down for some time. Take for example the coloumb electrostatic equation: F=Kq1q2/r^2.

I can understand why the force is inversly proportional to the distance, the further away you are from the source, the less you feel it. But why is it inversely proportional to the SECOND power of distance. Why not just to the first power or even third power? Or even, why not to the power of 2.31341, why does it have to be an integer?

Same thing can be asked with F=m*a. Why is it F linearly proportional to a and not perhaps to a squared?

However, I can understand that area of a square A = side^2. That is because we times one side with the other side. That makes sense, and also it has to be an integer (exactly 2). But I cant seem to find similar reasoning to the first two eqations above.

Any idea??? :cool: :smack:

The short stupid answer is that force is how we measure mass, so obviously they’re proportional, and the amount of force per total surface area at a distance r is constant so of course the surface area increases as r^2 (the surface area of a sphere of radius r is equal to 4pi x r^2), so the force must decrease as 1/r^2.

But such things have bedeviled other minds too, so you’re in good company. There have been a lot of theories that proposed that the “real” exponent was a little bit more or less than 2. We don’t really know why.

Actually, once you get away from the Coulomb law as the defining characteristic and look and the apropriate field-theoretical formuation, that’s pretty much exactly the answer. Force is (basically) proportional to Faraday field strength, and the Faraday field (as the exterior derivative of a 4-potential) is identically conserved. That is, the amount of field “passing through” any two surfaces surrounding the same source (charge) is the same. Double the radius, quadruple the surface area, quarter the field strength to get the same amount over four times the area.

Think of an electric charge producing an electric field that is uniform in all directions. Then the field will be uniform over the surface of a sphere centered on the charge the area of a sphere is proportional to the square of the radius so the field intensity on the surface of any sphere of radius r is proportional to 1/r[sup]2[/sup]

thanks, it makes sense now. The exponent of “2” is derived from the proportionality of the area of the sphere. :smiley:

Simple answer: that’s the definition of force: an acceleration applied to a mass. E.g., an rock dropped from 1 km from the surface of the Moon (no air resistance). The rock falls to the Moon’s surface at a constant accelleration, i.e., its velocity keeps increasing at a constant rate. What is creating this constant acceleration is the gravitational force between the Moon and the rock.

If we then move to another larger airless celestial body with twice the gravity of the Moon (at 1 km above), the same rock will fall with twice the acceleration.

As AWB said, that’s sort of the definition of force. It comes from Newton’s second law that says that acceleration is proportional to force and in the same direction.

I don’t know how Newton came up with that law. However, Kepler’s law relating time of an orbit to the distance from the primary, d[sup]3[/sup]/t[sup]2[/sup] = a constant can be derived from F = ma and the law of universal gravitation. Kepler’s law was found from observation of the planets.