center of mass?

Question

Pluto's diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km. Although the distance varies, they are often about 1.95×104 km apart, center-to-center.
Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

Answer 1

For center of mass distances D1 and D2 to bodies M1 and M2 are related as
M1*D1 = M2*D2
while D1 + D2 = D. Substitue D2 = D1*M1/M2 and get D1 + D1*M1/M2 = D1(M1 + M2)/M2 = D, so D1 = D*M2/(M1 + M2).
Because both planets have similar composition, they have similar density and M = ro*4/3*Pi*R^3. Substitute and get after simplifying:
D1 = D*R2^3/(R1^3 + R2^3)
D1 is distance to Pluto, so R1 is radius of Pluto and R2 is radius of Charon.
You can simplify calculations if you rewrite expression as
D1 = D/(1 + (R1/R2)^3)

Answer 2

The ratio of the distances [d's] of the centers of the masses is inversely proportional to the masses which in turn are proportional to the cube of the respective diameters.[ D"s]

d [P] / d [s] = m[s] /m[p] = D[s] ^3 / D [P]^3
[P]for Pluto and [s] for satellite:
d [P] / d [s] = [1250 / 2370] ^3 = 0.1467
d [P] / {d [P] +d [s]} = 0.1467 / 1.1467 = 0.1279
d[P] = 0.1279 * 1.95×10^4 =2494 km from Pluto's center,

Answer 3

Find an x such that
2370^3 * x = 1250^3 * (1.95*10^4 - x)
That's the distance the center of mass of the system is offset from Plutos center of mass. (That is, where they both have the same angular moment)

Doug

Answer 4

The key here is that, for a planet, the center of mass is the center of the planet. Thus, the mass of each planet can be considered a "point" mass, and the distance between these point masses would be (of course) the distance between planets plus the radii of each planet.

Jim, http://www.jimpettis.com/wheel/