Universal Gravitation help

First an inverse squared law is a big generalization. Basically all the inverse square law really means is that some quantity is proportional to 1/x^2 where x is some variable of interest. so the equation y = c/x^2 (c is some constant) is a general example of an inverse square law.

The masses are multipled because if we added them we would not have the correct dimensions. 1kg + 3kg = 4kg where as 1kg x 3kg = 3kg^2

As for the actual derivation I dont really know how he came up with it. I would have started assuming g is variable, thus...

F = mg
F/m = g
g = GMe/r^2

At this point I assume the mass and radius of the Earth had rough values. I think it could be reasoned that the mass should be on top because more mass leads to a greater weight force and that the radius should be on the bottom as increasing distance from a body yields a weaker gravitational force. Thus, g can be defined as follows

g = c*Me/r^n
where c is the constant of proportionality and n is the order of the radius

how these values were determined.....I have no idea, but thats a start

 

Essentially, Newton showed that this was the only force law that resulted in Kepler's laws for planetary motion.

 

Newton's constant G was first determined accurately, I think, by Cavendish, using a torsion balance.

 

Don't confuse G with g. Big G is the gravitational constant (6.673e-11 N m^2 / kg^2), and little g is acceleration due to gravity at the surface of the Earth (9.8 m/s^2).

[tex]F = \frac{GMm}{d^2}[/tex]

This solves for the gravitational force between them. Why do we multiply instead of add? Think of what would happen if we added. If a dust grain with a mass so close to 0 that you might as well call it 0 were on a bathroom scale, and we ADDED, the numerator of the equation would be enormous. But if we multiply, the numerator goes to 0, which is exactly what the bathroom scale would read.

Since
[tex]F = \frac{GMm}{d^2}[/tex]
and
[tex]F=ma[/tex]

you can conclude that

[tex]\frac{GMm}{d^2} = ma[/tex]

You've got a little m on each side of the equation, so they cancel:

[tex]a = \frac{GM}{d^2}[/tex]

giving you the acceleration formula.

 

You're approximately right that acceleration is change of velocity divided by time. Technically, that is average acceleration, but never mind for now.

When we say the acceleration due to gravity is, say, 9.8 m/s^2 near the surface of the Earth, we mean that, if you drop an object (e.g. a brick), its speed will change by 9.8 metres per second every second. So, if you drop it from rest, then after 1 second it will be travelling at 9.8 m/s. After 2 seconds, it will be travelling at 19.6 m/s, and so on.