Universal law of gravitational problem?

Question

the distance from the center of the earth to the center of the moon is 3.84 x 10^8m. Waht is the period of the moon as it orbits the earth? mass of the earth is 5.98 x 10^24kg. Work too please

Answer 1

T^2 = (4pi^2/GM)r^3

T = square root of (4pi^2/GM)r^3

T = square root of:

[(39.4784) / {(6.673x10^-11N.m^2/kg^2) (5.98 x 10^24kg)}] (3.84x10^8m)^3

T = square root of (56.02 x 10^11 s^2)

T = 2,366,858 seconds = 657.46 hours

T = 27.4 days

.

The moon phase cycle, seen from Earth, is a couple of days longer.

Answer 2

F = GmM/R^2 = mw^2R = C; where the force of gravity F equals the centrifugal force C because the moom m is at a fixed distance R from the center of earth M.  G = 6.67E-11 is the universal constant you can look up.  w = 2pi f = 2pi/T; where T = ? the period you are looking for.

Thus, GmM/R^2 = mw^2R and GmM = mw^2R^3; so that GM = w^2R^3 = (2pi/T)^2 R^3 = (4pi^2/T^2) R^3.  Then T^2 GM = 4pi^2 R^3; so that T^2 = 4pi^2 R^3/GM and T = 2pi sqrt(R^3/GM) = 2*3.14569*sqrt(((3.84E8)^3)/((6.67E-11)*... = 27.39972371 days, which is called a Lunar month.

The physics is this, the gravitational pull on the Moon is offset by its centrifugal force.  This is why it stays there in orbit rather than falling in on us and ruining our whole day.