# The moon orbits the earth at a distance of 3.85x10^8 m. The moon orbits the earth at a distance of 3.85x10^8 m. Assume that this distance is between the centers of the earth and the moon and that...

The moon orbits the earth at a distance of 3.85x10^8 m.

The moon orbits the earth at a distance of 3.85x10^8 m. Assume that this distance is between the centers of the earth and the moon and that the mass of the earth is 5.98x10^24 kg. Find the period for the moon's motion around the earth. Express the answer in days and compare it to the lenght of a month.

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### 1 Answer

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.*

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