You have been hired as a consultant for the new Star Trek TV series to make sure that any science on the show is correct. In this episode, the crew of the Enterprise discovers an abandoned space station in deep space far from any stars. This station is obviously the work of an advanced race and consists of four 4 x 10²⁰ kg asteroids configured so that each is at the corner of a square with 150 km sides. According to the tricorder, the station has been abandoned for at least two centuries. You know that such a configuration is unstable and worry whether there would be observable motion of the asteroids after two hundred years so you 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.

Note: Far away from Earth, we can't use a constant gravitational force Fg=mg. Instead, we need to use a more general expression: F_g = (Gm₁m₂)/r², where mi and m₂ are the objects masses, r is the distance between them, and G is a constant, G = 6.674 x 10^-11N•m²/kg².

Your answer must be organized, clear, and include the elements of the problem solving path:

1. Focus the Problem

2. Describe the Physics

3. Plan the Solution

4. Execute and Evaluate

Problem Solving Path-

1. Focus the Problem-

a. Re-state the problem in your own words.

b. Draw a diagram(s).

c. Label the pans of the diagram that have physical significance to this problem.

d. Identify the physical quantity which it is the object of this problem to find.

2. Describe the Physics-

a. List all the simple physical quantities relevant to this problem (known's) and give values.

b. List the computable physical quantities you will need to solve the problem (unknowns).

3. Plan the Solution-

a. List and describe the relational, physical principles, concepts, and equations that are needed to solve the problem.

b. Arrange them in the order you will use them and explain with words how each is to be used.

c. Refer to your diagram for context.

d. If you find any missing knowns or unknowns along the way, add them to your information for step 2.

4. Execute the Solution and Evaluate-

a. Use information from step 2 in your plan described in step 3 to find any intermediate values needed.

b. Use information from step 2 in your plan from step 3 to find a value for the physical quantity (or quantities) that is the final goal.

c. Check the units and directions to make sure they are consistent.

d. Check to see if the answer is physically possible.

e. Answer any qualitative questions from the problem statement based on your physical solution.