Two satellities `1 and 2` orbiting with the time periods `T_1 and T_2`, respectively, lie on the same line as shown in (Fig. 5.167). After what minimum time, again the satellities will remain on the same line ? Assume that the two satellities should lie in same side of the center of their concentric circular paths.
.

Explanation

Let the satellities `1 and 2` describe angles `theta_1 and theta_2` at the center of earth during a time `t`, respectively. Suppose `2` is ahead of by `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'` sat. For this `theta = 2 pi`. This gives `theta_2 - theta_1 = 2 pi`. Dividing `T'` in both sides,
we have `(theta_2)/(T') - (theta_1)/(T') = (2 pi)/(T')`
Substituting `(theta_2)/(T') = omega_2 and (theta_1)/(T') = omega_1`, we have
`(2 pi)/(T') = omega_2 - omega_1` where `omega_2 = (2 pi)/(T_2)` and `omega_1 = (2 pi)/(T_1)`.
Then, `(1)/(T') = (1)/(T_2) - (1)/(T_1)`
This gives `T' = (T_1 T_2)/(T_1 - T_2)`