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Keplers Third Law

Step-by-step explanation:

Kepler's Third Law relates the period of an orbit to the radius of an orbit, if the orbit is circular, and to the semimajor axis if the orbit is elliptical.

Let's derive it for a circular orbit, assuming a mass m is orbiting a mass M, with r being the radius of the orbit.

ΣF = ma

GmM/r2 = mv2/r

Cancelling the m's and a factor of r gives:

GM/r = v2

We can bring in the period using:

v = 2πr / T

This gives GM/r = 4π2r2/T2

Re-arranging gives:

Kepler's Third Law: T2 = (4π2/GM) r3

For an system like the solar system, M is the mass of the Sun. So the constant in the brackets is the same for every planet, and we get the relationship that the period of the orbit is proportional to r3/2.

The same with satellites orbiting the Earth - in that case M is the mass of the Earth, but again the larger the orbit the larger the period.

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