Gravitation and the Principle of Superposition

[peaceandlove]

Homework Statement

A square of edge length 19.0 cm is formed by four spheres of masses m1 = 5.10 g, m2 = 3.00 g, m3 = 1.20 g, and m4 = 5.10 g. In unit-vector notation, what is the net gravitational force from them on a central sphere with mass m5 = 3.00 g?

Homework Equations

G*m_1*m_2/d^2

The Attempt at a Solution

First, I calculated the distance, d, use the edge length they give you and the pythagorean theorem to solve for a diagonal in the square. I divided this answer by two for the distance between the center particle and any of the 4 particles at the extremities.

Next, I ran the formula above for each of the 4 particles to solve for the forces they exert. However, I'm confused as how to take the difference between the opposing forces to get two net forces, and calculate the forces in the x and y directions by using trig. For example, I'm unsure whether to subtract F_54 from F_51 or vice versa?

 

[LowlyPion]

You will want to add them as vectors.

So draw each of them, and then account for the x and y components of each.

As a short cut you can take the difference between forces from opposite corners, since they necessarily are opposite in direction at the center point.

 

[peaceandlove]

You will want to add them as vectors.

So draw each of them, and then account for the x and y components of each.

As a short cut you can take the difference between forces from opposite corners, since they necessarily are opposite in direction at the center point.

For the shortcut, do I subtract F_54 (top left) from F_51 (bottom right) or vice versa? F_53 (bottom left) from F_52 (top right) or vice verse?

 

[LowlyPion]

For the shortcut, do I subtract F_54 (top left) from F_51 (bottom right) or vice versa? F_53 (bottom left) from F_52 (top right) or vice verse?

The ones that are in opposite directions you can, both pairs, then add the those results as vectors.

 

[peaceandlove]

F_1= -3.997836793732E-14 (x-coordinate) & 3.997836793732E-14 (y-coordinate)
F_2= 2.351668702195E-14 (x-coordinate) & 2.351668702195E-14 (y-coordinate)
F_3= -9.406674808782E-15 (x-coordinate) & -9.406674808782E-15 (y-coordinate)
F_4= 3.997836793732E-14 (x-coordinate) & -3.997836793732E-14 (y-coordinate)

Added all together: x-coordinate=1.41103e-14 and y-coordinate=1.41103e-14.