**Problem**

Derive Keplar’s third law of planetary orbits for comparing the periods and orbital radius of two planets. This law is sometimes called the harmonic law.

Assume the orbits are circles.

**Solution**

The net force in orbital motion is the gravitational force

- .

Since the net force is the centripetal force for planetary orbits,

- .

For a planet with an orbital radius

- .

For a planet with an orbital radius

- .

Divide the two equations to discover that

- .

**Note**

Most planetary orbits are elliptical, not circular. Although the derivation uses more advanced mathematics, the final result is not that different.

Here, is the semi-major axis of the ellipse. Thus the Harmonic law would be

- .

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