Gravitation with multiple objects by using center of mass

[XSethX]

Four 9.5 kg spheres are located at the corners of a square of side .60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere by the other three.

So, essentially my question is, why can I not use center of mass of the other 3 spheres to calculate the gravitational force? I set the top left sphere as my frame of reference, and found the center of mass of the bottom left and top right spheres and calculated the gravitational force at that center of mass on the top left sphere (I narrowed down my error to this part.) Why can I not do this? Thanks in advance!

Edit: Very very sorry, realized I should have posted this in "Introductory Physics Homework"...

 

[Isaac0427]

I'm not sure exactly why you can't do it, but I also don't see why you would want to do it. I would calculate the force of gravity from each sphere and add the vectors.

 

[Isaac0427]

By the way, welcome to PF!

 

[Drakkith]

I set the top left sphere as my frame of reference, and found the center of mass of the bottom left and top right spheres and calculated the gravitational force at that center of mass on the top left sphere (I narrowed down my error to this part.)

What happened to the bottom right sphere?

 

[Nathanael]

It won't generally work out if you use center of mass to calculate the resultant gravitational force.
(It is fortunate that it turns out true for planets and such.)
To see why it doesn't make sense as a general rule, consider the following question:
What is the gravitational force at the origin from two identical particles whose positions along some axis are -L+dL and L+dL? We expect (by symmetry) that as we choose smaller dL, the gravitational force should go to zero. Notice though, that the center of mass is a distance dL from the origin; so using your method the gravitational force gets arbitrarily large (we are dividing by dL squared). Clearly incorrect.

 

[XSethX]

What happened to the bottom right sphere?

I didn't forget, I just calculated that after and added to the number I found. Both times I tried to solve the problem, that value was the same, so I figured my error was with using the center of mass for the bottom left and top right spheres. Thanks!

 

[XSethX]

I'm not sure exactly why you can't do it, but I also don't see why you would want to do it. I would calculate the force of gravity from each sphere and add the vectors.

Yeah, that's how I got the right answer. I was just curious about why using center of mass didn't work. Thanks, though!

 

[XSethX]

It won't generally work out if you use center of mass to calculate the resultant gravitational force.
(It is fortunate that it turns out true for planets and such.)
To see why it doesn't make sense as a general rule, consider the following question:
What is the gravitational force at the origin from two identical particles whose positions along some axis are -L+dL and L+dL? We expect (by symmetry) that as we choose smaller dL, the gravitational force should go to zero. Notice though, that the center of mass is a distance dL from the origin; so using your method the gravitational force gets arbitrarily large (we are dividing by dL squared). Clearly incorrect.

Oh, I understand now. Thank you!