3D tension problem (static equilibrium)

In summary: It states that in a regular tetrahedron, each angle between any two edges is 70.5 degrees. Using this information, we can set up a system of equations to solve for the tensions T1, T2, and T3. Since T1=T2=T3, we can set them all equal to T and simplify the equations. This gives us T=50N/3. So the magnitude of the tension in each cable is 50N/3. This thought process can be applied to solve problems involving other regular polyhedrons, where symmetry can be used to simplify the problem. In cases where there isn't symmetry, we would need to use other methods such as vector decomposition to solve for the tensions.
  • #1
silentskills
1
0

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.
 
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  • #2
silentskills said:

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 .
 

1. What is a 3D tension problem in static equilibrium?

A 3D tension problem in static equilibrium is a type of physics problem that involves calculating the forces acting on an object in three-dimensional space. These forces include tension, which is a pulling force, and the object is assumed to be at rest or in a state of equilibrium.

2. How do you solve a 3D tension problem in static equilibrium?

To solve a 3D tension problem in static equilibrium, you need to first draw a free body diagram of the object and identify all the forces acting on it. Then, you can use the equations of static equilibrium to calculate the unknown forces and determine if the object is in a state of equilibrium.

3. What are the equations of static equilibrium?

The equations of static equilibrium are the conditions that must be satisfied for an object to be in a state of equilibrium. These include the sum of all the forces in the x, y, and z directions being equal to zero, as well as the sum of all the torques (rotational forces) being equal to zero.

4. What factors can affect the tension in a 3D tension problem?

The tension in a 3D tension problem can be affected by several factors, including the weight of the object, the angle of the ropes or cables, and the presence of other forces acting on the object. The tension can also change if the object is in motion or if the ropes or cables are stretched or compressed.

5. Why is solving 3D tension problems in static equilibrium important?

Solving 3D tension problems in static equilibrium is important because it allows us to understand and predict the behavior of objects under different forces. This is particularly useful in engineering and construction, where structures need to be designed to withstand various forces and maintain stability.

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