Gravitational pull, superposition

[lemonpie]

Homework Statement

Two spheres of mass m and a third sphere of mass M form an equilateral triangle, and a fourth sphere of mass m4 is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is M in terms of m? (b) If we double the value of m4, what then is the magnitude of the net gravitational force on the central sphere?

Homework Equations

Fnet = F1,4 + F2,4 + F3,4
F = GMm/r²

The Attempt at a Solution

F1,4 = GMm/r²
F2,4 =Gm²/r²
F3,4 =Gm²/r²

Fnet = 0 = F1,4 + F2,4 + F3,4
0 = GMm/r² + Gm²/r² +Gm²/r²
0 = Mm + 2m²
Mm = -2m²
M = -2m

this is not correct. please help me figure out what I'm doing wrong. thanks.

 

[D H]

Hint: You are not using m4 in any of your force calculations.

 

[lemonpie]

oh, for some reason i thought it said m = m4.

Fnet = 0 = F1,4 + F2,4 + F3,4
0 = GMm4/r2 + Gmm4/r2 + Gmm4/r2
0 = M + 2m (canceling out G, m4, and r2)
M = -2m

um...

 

[D H]

Gravitational force, like all forces, is a vector. You are treating it as a scalar.

 

[lemonpie]

i understand that it's a vector, but where do i set the origin? if i set it at m4, i still don't get the answer that i want. i get M = 2m, which is still not correct.

 

[D H]

Changing the origin or using a different set of axes will not change the answer -- if you do your work correctly. Think of it this way: Does astronomers' decision regarding the directions in the x, y, z axes point, or the point they call the origin magically change the Sun's gravitational force on the Earth?

 

[lemonpie]

okay... so what am i doing wrong?

 

[Doc Al]

What you care about is the direction of the gravitational force contributions from each of the three mass. To orient yourself, try this: Let the two m masses be along the x-axis, on either side of the y-axis; Let the M mass be on the positive y-axis. So where would m4 be in this picture? Which way will the force vector from M point? Which way will the force vectors from the two other masses point? (Use what you know about equilateral triangles.)

Take advantage of symmetry and find the components of those force vectors.