Pluto's diameter is approximately 2370 , and the diameter of its satellite Charon is 1250 . Although the dista

Pluto's diameter is approximately 2370km , and the diameter of its satellite Charon is 1250km . Although the distance varies, they are often about 1.97×104km apart, centre-to-centre.

Part A
Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the centre of mass of this system relative to the centre of Pluto.(answe in km)

Mass = volume / density .. being of equal density, M ∝ diameter(D)³
Mp = k.Dp³ .. and Mc = k.Dc³

Let CoM point be R from centre of Pluto .. the system is in 'balance' about this point ('moments' equal either side of R)

Mp x R = Mc x (1.97^4 - R) .. .. R (Mp + Mc) = 1.97^4 Mc

R = 1.97^4 Mc / (Mp + Mc)

R = 1.97^4 x k.Dc³ / k(Dp³ + Dc³) .. .. k cancels out

R = 1.97^4 x 1250³ / (2370³ + 1250³) .. .. ►R = 2521 km

Let the density of both planets be ρ.

V = 4/3 * pi * r^3
and
M = ρ * V
So the mass of Pluto is:
Mp = ρ * 4/3 * pi * (1/2 * 2370)^3
The mass of Charon in:
Mc = ρ * 4/3 * pi * (1/2 * 1250)^3

Let the position of Pluto be x1. Set x1 = 0 for simplicity
Let the position of Charon be x2. So x2 = 1.97e4 km
Let the position of the centre of mass be x. Then,

Mp * x1 + Mc * x2 = (Mp + Mc) * x
So x = Mc * x2 / [ Mp + Mc ]
x = 2.52e3 m

This means the centre of mass is located slightly above the surface of Pluto.