The exact mass of the moon was not determined until the Apollo lunar orbiter went into orbit around the moon. They used the period and the orbital radius to find the mass. But since you already know the mass of the moon, if the spaceship was orbiting 485 km above the surface of the moon, how long did it take for the lunar orbiter to make one complete orbit?
Set Gravitational force equal to centripetal force:
Gm1m2/r^2 = m2v^2/r
Notice m2 crosses out (as well as one of the r's)
solve for v.
Now find the circumference of the orbit (2pi r) and divide my the velocity.
Et voila.
To determine the time it takes for the lunar orbiter to make one complete orbit, we can use Kepler's Third Law of planetary motion, which relates the orbital period of a satellite to its orbital radius and the mass of the central body it is orbiting.
Kepler's Third Law states that the square of the orbital period (T) is proportional to the cube of the orbital radius (r) of the satellite:
T^2 = (4π^2/GM) * r^3
Where:
T is the orbital period
r is the orbital radius
G is the gravitational constant
M is the mass of the central body (in this case, the moon)
Given that the spacecraft is orbiting 485 km above the surface of the moon, the orbital radius (r) would be the sum of the radius of the moon and the altitude of the orbit. The average radius of the moon is approximately 1,737.1 km, so the total radius (R) would be:
R = 1,737.1 km + 485 km = 2,222.1 km
Now, we can rearrange Kepler's Third Law equation to solve for the orbital period (T):
T = √((4π^2/GM) * r^3)
To find the orbital period, we need the mass of the moon. According to your statement, we have already determined the mass of the moon. Let's assume it is denoted as M_moon.
Plugging in the known values:
T = √((4π^2/G * M_moon) * R^3)
We would need the exact mass of the moon to calculate the orbital period accurately. Once you provide the mass of the moon, we can use this formula to determine the time it takes for the lunar orbiter to make one complete orbit.