Question

Two satellites $1$ and $2$ orbiting with the time periods $T_1$ and $T_2$, respectively, lie on the same line as shown in Fig. After what minimum time, again the satellites will remain on the same line? Assume that the two satellites should lie in same side of the centre of their concentric circular paths.

Answer 1

Let the satellites $1$ and $2$ described angle ${\theta }_1$ and ${\theta }_2$ at the center of earth during a time $t$ respectively .Suppose $2$ is ahead of $1$ by an angle $\theta$ hence we can write

$\theta -{\theta }_2-{\theta }_1$

They will be again lie on same line after a minimum time $T$ say.for this $\theta =2\pi$.This gives ${\theta }_2-{\theta }_1=2\pi$Dividing $T$ in both sides we have $\frac{{\theta }_2}{T^{\prime }}-\frac{{\theta }_1}{T^{\prime }}=\frac{2\pi }{T^{\prime }}$

Substituting $\frac{{\theta }_2}{T^{\prime }}={\omega }_2$ and $\frac{{\theta }_1}{T^{\prime }}={\omega }_1$,we have

$\frac{2\pi }{T^{\prime }}={\omega }_2-{\omega }_1$ where ${\omega }_2=\frac{2\pi }{T_2}$ and ${\omega }_1=\frac{2\pi }{T_1}$

Then $\frac{1}{T^{\prime }}=\frac{1}{T_2}-\frac{1}{T_1}$

This gives $T^{\prime }=\frac{T_1{T}_2}{T_1-T_2}$