3D tension problem (static equilibrium)

[silentskills]

Homework Statement

A block of weight 50N is hung by 3 cables from the ceiling. Each rope ZA, ZB, ZC converges at Z so that they form a tetrahedron. (ZA=ZB=ZC=AB=BC=CA). Find the magnitude of the tension of each cable.

a=0
tetrahedron symmetry

Homework Equations

F=mg

The Attempt at a Solution

I realized that T1=T2=T3 due to the symmetrical nature of tetrahedrons. The angles weren't given but each side of a tetrahedron is composed of 3x60 degree angles.

My approach was to represent the problem in 2D since there were some obvious vector symmetries in the x,y plane.
I proceeded to decompose the F(x,y) components, which cancel each other out. Can be proven with a simple qualitative 2D vector sum, even though the quantities weren't given.
I then tried: F(z) = T1 sin + T2 sinβ + T3 sinβ = 50N.
or
(3)(T)(sinβ) = 50N

But then I couldn't figure out how to find β. Initially I assumed it was 60 degrees, but after checking my work the math didn't work out.
I'm basically curious as to what is the thought process required to solve a problem like this, or a more general case where there aren't any nice symmetric features.

 

[SammyS]

Homework Statement

A block of weight 50N is hung by 3 cables from the ceiling. Each rope ZA, ZB, ZC converges at Z so that they form a tetrahedron. (ZA=ZB=ZC=AB=BC=CA). Find the magnitude of the tension of each cable.

a=0
tetrahedron symmetry

Homework Equations

F=mg

The Attempt at a Solution

I realized that T1=T2=T3 due to the symmetrical nature of tetrahedrons. The angles weren't given but each side of a tetrahedron is composed of 3x60 degree angles.

My approach was to represent the problem in 2D since there were some obvious vector symmetries in the x,y plane.
I proceeded to decompose the F(x,y) components, which cancel each other out. Can be proven with a simple qualitative 2D vector sum, even though the quantities weren't given.
I then tried: F(z) = T1 sin + T2 sinβ + T3 sinβ = 50N.
or
(3)(T)(sinβ) = 50N

But then I couldn't figure out how to find β. Initially I assumed it was 60 degrees, but after checking my work the math didn't work out.
I'm basically curious as to what is the thought process required to solve a problem like this, or a more general case where there aren't any nice symmetric features.

You could check out the Wikipedia article on tetrahedrons .