Here's the question:
As space colonization expands, it's important to build new stations from local materials instead of bringing everything from Earth. Your latest task (besides asking for a raise) is to check the long-term stability of a proposed configuration of asteroids for a deep-space station, far from any stars. The schematic you receive has four 4 x 10^20 kg asteroids configured so that each is at the corner of a square with 150 km sides. The asteroids will gravitationally attract each other, but the designer claims it will be stable for hundreds of years. To check, you want to calculate the acceleration of one of the asteroids in the proposed configuration. Make sure you give both the magnitude and the direction of the acceleration.
Hint: Consider just one of the asteroids and analyze the forces it will feel from the other three.
Note: Far away from Earth, we can't use a constant gravitational force Fg = mg. Instead, we need to use a more general expression: Fg = (Gm1m2)/r^2 , where m1 and m2 are two objects' masses, r is the distance between them, and G is a constant, G = (6.674 x 10^-11)N*m^2/kg^2
Fg = (G*m1m2)/r^2
The Attempt at a Solution
All I have done is start the free body diagram of one of the asteroids:
Can I find the resultant of F m3->m1 and F m2->m1 and add that to F m4->m1 to find the total Fg towards the center?
I'm stuck. Any tips on getting started?
√(752 + 752) = 106.07km
asteroid has to travel 106.07km to get to the center.