The moon is rotating around earth. So in the moon there is a centrifugal force acting towards earth.
`F = mv^2/r`
m = mass of moon
r = distance between moon and earth
v = velocity of moon
Earth exerts a gravitational force on earth as given by Newtons low of gravity.
`F = G(Mm)/r^2`
M = mass of earth
G = Newtonian gravity constant
But by both these equations it expresses the same thing.
`GMm/r^2 = mv^2/r`
` v = sqrt((GM)/r)`
So the linear velocity of moon is 1020m/s.
For one round around earth the moon travels `2pir` distance.
This distance is equal to time*velocity of moon.
`2*pi*3.85x10^8 = 1020*t`
`t = 2.38xx10^6s`
Number of days `= (2.38xx10^6)/(60xx60xx24) = 27.5`
So it will take 27.5 days to complete full motion around earth.