Gravitational field at the fourth corner of a square

[blackheart]

1. Three objects of equal mass are located at three corners of a square of edge length d. Find the magnitude and direction of the gravitational field at the fourth corner due to these objects. (Use the following as necessary: G, d, and m.)

2. Homework Equations :

g = Gm/r²

3. My attempt at the solution: I labeled the masses as 1, 2, and 3, where 1 and 3 are adjacent to the fourth corner and 2 is in the opposite corner from the fourth.

g₁ = g₃ = Gm/d²
g₂ = Gm/(d$$\sqrt{}2$$²)

g_y = g₁ + g_2y

= Gm/d² + g₂ sin 45

= Gm/d² + Gm/(d$$\sqrt{}2$$)²

g_x = g₃ + g_2x

= Gm/d² + g₂ cos 45

= Gm/d² + Gm(d$$\sqrt{}2$$)²

g² = g_x² + g_y²

g = $$\sqrt{}(2Gm/d^{2 + Gm/d^2 cos 45}[/SUP])$$

 

[issacnewton]

your calculations for the x and y components of g are wrong. use the correct values of
sin and cos

$$\sin(45)=\cos(45)=\frac{1}{\sqrt{2}}$$

reworkout that