Question

In this problem, you will calculate the location of the center of mass for the Earth-Moon system, and then you will calculate the center of mass of the earth-Moon-Sun system. The mass of the Moon is \(7.35 \times 10^{22} \mathrm{~kg}\), the mass of the Earth is \(6.00 \times 10^{24} \mathrm{~kg}\), and the mass of the sun is \(2.00 \times 10^{30} \mathrm{~kg}\). The distance between the Moon and the Earth is \(3.80 \times 10^{5} \mathrm{~km}\). The distance between the Earth and the Sun is \(1.50 \times 10^{8} \mathrm{~km}\)

Calculate the location \(x_{\mathrm{cm}}\) of the center of mass ofthe Earth-Moon system. Use a coordinate system in which the center of the Earth is at \(x=0\) and the Moon is located in the positive \(x\) direction.

Answer 1

Let m1 = 6.00*10^24 kg, x1 = 0
m2 = 7.35*10^22 kg, x2 = 3.80*10^5 km

the location of the center of mass of the Earth-Moon system,

Xcm = (m1*x1 + m2*x2)/(m1 + m2)

= (6*10^24*0 + 7.35*10^22*3.8*10^5)/(6*10^24 + 7.35*10^22)

= 4599 km

= 4.60*10^3 km

to find center of mass of Earth-Moon-Sun system

m1 = 2*10^30 kg, x1 = 0
m2 = 6*10^24 kg, x2 = 1.50*10^8 km
m3 = 7.35*10^22 kg, x3 = (1.5*10^8 + 3.8*10^5) km

the location of the center of mass of the Earth-Moon-Sun system,

Xcm = (m1*x1 + m2*x2 + m3*x3)/(m1 + m2 + m3)

= (2*10^30*0 + 6*10^24*1.5*10^8 + 7.35*10^22*(1.5*10^8 + 3.8*10^5) )/(2*10^30 + 6*10^24 + 7.35*10^22)

= 455 km (from the center of the sun)