Unravelling Kepler's Law: How F = m/r^2 is Reached

[Miike012]

what is it exactly?

and in my book they had an equation

F = (4)(pi)^2(m)(r)/T^2

then they said by using keplers law... they arrived to a new equation that relates the gravitational force exerted by the sun which is...
F = m/r^2

If Keplers law says T = r^3/2 how in the heck did they go from F = (4)(pi)^2(m)(r)/T^2
to F = m/r^2 by substituting T for r?
Where did (4)(pi)^2(r) go ?

 

[gneill]

It would appear that they started with the centripetal force:
[tex] F = m\frac{v^2}{r}~~~~~\text{where:}~~~~v = \frac{2 \pi r}{T}[/tex]
[tex] F = \frac{4 \pi^2 m r}{T^2}[/tex]
Now, Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For a circular orbit we identify the semi-major axis with the orbital radius. Note that the law as stated does not give us the constant of proportionality, so we write: [itex] r^3 \propto T^2[/itex]. So we can replace the T² in the force equation with r³ but without a precise constant of proportionality it doesn't make sense to retain the others, so that:
[tex] F \propto \frac{4 \pi^2 m r}{r^3} \propto \frac{m}{r^2}[/tex]

 

[NascentOxygen]

For different planets, Kepler's third Law says (R^3)/(T^2) = constant
and equals unity when using units of years and astronomical units (A.U.).

 

[Miike012]

Ok so I see that I r will divide out and it will equal 4(pi)^2m/r^2... where did the 4 pi^2 go?

 

[asteriatic]

Kepler's Law states that the square of the orbital period of a planet is directly proportional to the cube of its average distance from the sun. Mathematically, this can be expressed as T^2 ∝ r^3.

In the first equation, F = (4)(pi)^2(m)(r)/T^2, the term (4)(pi)^2(r) is simply a constant that represents the universal gravitational constant, G, and the orbital period, T.

By substituting T for r in the equation, we can rearrange it to get F = m/r^2, which is the familiar form of Newton's Law of Universal Gravitation. This equation shows that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

So, in essence, by using Kepler's Law and substituting T for r, we are able to derive Newton's Law of Universal Gravitation, which is a fundamental equation in physics that explains the force of gravity between objects.