Universal gravitation 3- determine the mass of the Earth

[dani123]

Homework Statement

The moon orbits the Earth at a distance of 3.84x10⁸m from the centre of Earth. The moon has a period of about 27.3 days. From these values, determine the mass of the Earth.

Homework Equations

Kepler's 3rd law: (T_a/T_b)²=(R_a/R_b)³

motion of planets must conform to circular motion equation: F_c=4∏²mR/T²

From Kepler's 3rd law: R³/T²=K or T²=R³/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏²K_s)*m/R²=Gm_sm/R²

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏²K_e)m/R²=Gm_em/R²

Newton's Universal Law of Gravitation: F=Gm₁m₂/d²

value of universal gravitation constant is: G=6.67x10^-11N*m²/kg²

weight of object on or near Earth: weight=F_g=m_og, where g=9.8 N/kg
F_g=Gm_om_e/R_e²

g=Gm_e/(R_e)²

determine the mass of the Earth: m_e=g(R_e)²/G

speed of satellite as it orbits the Earth: v=√GM_e/R, where R=R_e+h

period of the Earth-orbiting satellite: T=2∏√R³/GM_e

Field strength in units N/kg: g=F/m

Determine mass of planet when given orbital period and mean orbital radius: M_p=4∏²R_p³/GT_p²

The Attempt at a Solution

So used m_E=g(R_E)²/G and i was confused as to which value to use for R_E... do I use the Earth's radius or do I use the distance from the centre of the Earth to moon that is given in the problem...

If i use the value they give in the problem and g=9.8 I would obtain
m_E=2.167x10²⁸kg

Does this seem right? If someone could correct me if I am wrong here, that would be greatly appreciated... thanks so much in advance :)

 

[MalachiK]

Let's see, I think it's proably okay to assume that the moon is moving in a circle around the Earth at a constant speed. If that's true, do you know any formulae for the centripetal force that must be acting on a mass doing uniform circular motion? Wouldn't that be exactly equal to the size of the gravitational force between the Earth and the moon?