Question

Two planets undertake uniform circular motion around a star. For the two planets, the mass, speed and distance from the star's centre is respectively m,v, r and m',v', r'. If r'=4r , the ratio T'/T of the periods of revolution of the planets is? The answer is 8 however I am not sure which formula to use because the ones I have used do not give me that answer.

Answer 1

According to the third Kepler's law, the periods of revolution are in the following relation with distances from the star:

$$\dfrac{T'^2}{T^2} = \dfrac{r'^3}{r^3}$$

If we substitute $r' = 4r$ and express the ratio $T'/T$ we will obtain:

$$\dfrac{T'}{T} = \sqrt{\dfrac{(4r)^3}{r^3}} = \sqrt{4^3} = 8.$$

Thus, your answer is correct according to the third Kepler's law.

Answer. 8.