What is the Center of Mass of the Earth + Satellite System?

[smpolisetti]

This problem is driving me crazy!

Homework Statement

How far from the center of the Earth (in km) is the center of mass of the Earth + satellite system if the satellite has a mass of 2970 kg and is in orbit 42400 km above the surface of the Earth. The mass of the Earth is 5.98*10^24 and the radius of the Earth is 6370

So:
Mass of earth: 5.98*10^24 kg
Mass of satellite: 2960 kg
Distance of satellite from surface of the earth: 42400
Radius of earth: 6370

Homework Equations

Center of mass = (x1m1 + x2m2)/ (m1+m2)
m= mass
x= distance

The Attempt at a Solution

Here was my attempt

I set up center of mass = [(0km (5.98*10^24)) + (2970kg * 42400km) ] / ((5.98*10^24)+2970)

From that I got 2.106*10-17 km

Which I then added to the radius of the Earth to get 6370 km

But that's wrong, what was my mistake?

 

[cepheid]

I set up center of mass = [(0km (5.98*10^24)) + (2970kg * 42400km) ] / ((5.98*10^24)+2970)

All distances in the problem are being measured relative to the centre of the Earth. So, this distance in red should be the distance between the satellite and the centre of the Earth.

From that I got 2.106*10-17 km

Which I then added to the radius of the Earth to get 6370 km

Why would you do this? All distances in this problem are referenced to the centre of the Earth. That means that the answer you get from your equation will be the position of the centre of mass relative to the centre of the Earth, which is exactly what you want. No corrections are necessary.

Also, please THINK about your answers and whether they make sense. Use your intuition. If the satellite weren't there, where would the centre of mass of the Earth be? It would be at the centre of the Earth, right? So, is adding a satellite, an object whose mass is negligible compared to the planet, really going to shift the centre of mass very far away from the centre of the planet?

 

[smpolisetti]

Thanks a ton!