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.

Therefore;

`GMm/r^2 = mv^2/r`

        ` v = sqrt((GM)/r)`

          `= sqrt(6.67xx10^(-11)*(5.98xx10^24)/3.85xx10^8)`

          `= 1.02xx10^3`

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.