A satellite is placed between the Earth and the Moon, along a straight line that connects their centers of mass. The satellite has an orbital period around the Earth that is the same as that of the Moon, 27.3 days. How far away from the Earth should this satellite be placed?
To determine the distance the satellite should be placed from the Earth, we need to consider the gravitational forces acting on the satellite by the Earth and the Moon.
First, let's find the orbital radius of the Moon. We know that the Moon completes one orbit around the Earth in 27.3 days. The period of an orbit, T, is related to the radius of the orbit, R, by the equation:
T^2 ∝ R^3
So, if the orbital period of the Moon is 27.3 days (or 27.3 * 24 * 60 * 60 seconds), then we can calculate the orbital radius of the Moon. Let's call it R_moon.
Now, using the same equation, we can find the required orbital radius for our satellite by equating the periods:
T_moon^2 = T_satellite^2
R_moon^3 = R_satellite^3
Since we know the value of R_moon and want to find R_satellite, we can rearrange the equation:
R_satellite = R_moon * (T_satellite / T_moon)^(2/3)
Substituting the known values, we have:
R_satellite = R_moon * (27.3 days / 27.3 days)^(2/3)
Simplifying, we get:
R_satellite = R_moon
Hence, the satellite should be placed at the same distance as the Moon from the Earth.