Lunar orbit question (calculating the mass ratio of earth and moon)

[jkena04]

The problem I am struggling with is this . .

The centre of the Earth-Moon system is 4.7x10^3 km from the centre of the earth. Calculate;

i.the ratio of the mass of the Earth with the mass of the moon.
ii.The acceleration of free fall on the Moon's surface.

Constants
Distance between centre of mass of the Earth and centre of mass of moon - 3.8x10^5 km
Radius of Earth - 6.4x10^3 km
radius of moon - 1.7x10^3 km

Im a bit confused over how the moon actually orbits the earth. Do both planets orbit a centre point? If this is the case then for the Earth is in circular motion about a centre point, hence;

F(centripetal)=(m(earth)*v^2)/r = Gravitational force = (G*m(earth)*m(moon))/R^2

where R=distance between Earth and moon
r=radius of circular orbit

is this correct? also do the Earth and the moon also have the same tangential or angular speed?

Thank you in advance for your help!

 

[gneill]

You could go the route of balancing the centripetal forces as you began to do. If so, you might want to cast it in terms of angular velocity rather than tangential velocity, since the angular velocities of the Earth and Moon about their mutual center of gravity must be equal (and independent of their orbital radii about the center of mass).

Alternatively, suppose you were asked to find the center of mass (COM) of two stationary objects sitting in front of you. You're given their masses, m1 and m2, and the total distance between them, d, and you need to find the distance of the COM from the center of m1 in terms of the masses and distances given. How would you proceed?