Length of Simple Elastic Rods in Equilibrium

[tsuinami]

Homework Statement

I have a simple structure made up of joints and rods. Please feel free to check out the diagram here:

http://rsizr.com/etc/hardelastics.pdf"

The rigid rods do not deform, and the elastic rods will deform only in the horizontal dimension. If all five elastic rods are the same material, same rest length (length = 1) and only respond to axial forces at the endpoints, what will be their equilibrium lengths given the structure?

Please ignore gravity, friction, or anything else unmentioned. Thanks very much for your help!

2. The attempt at a solution

I came up with a solution to an easier version of this problem:

http://rsizr.com/etc/elasticsprob.pdf"

Through simple introspection, using these equations:

x+y = 5
y+z = 3
y=x+z

solution: x = 7/3, y = 8/3, z = 1/3

This solution seems to make sense but I am also not sure because it's not grounded with real physics equations :)

(I tried to apply equations for elastic equilibrium but got lost...)

 

[anathema]

Thank you for your post and for sharing your solution to the problem. I would like to offer some insights and suggestions on this problem.

Firstly, your solution for the simpler version of the problem seems reasonable, but as you mentioned, it is not grounded in real physics equations. In order to accurately solve this problem, we need to consider the material properties of the elastic rods and the forces acting on them.

To start, we can assume that the elastic rods have linear elastic behavior, meaning that the deformation is directly proportional to the applied force. This can be described by Hooke's Law, which states that the force (F) applied to an elastic rod is equal to the product of its spring constant (k) and the change in its length (ΔL): F = kΔL.

Next, we need to consider the equilibrium conditions for the structure. In this case, we can assume that the structure is in static equilibrium, meaning that the forces acting on it are balanced. This can be described by Newton's Second Law, which states that the sum of the forces (ΣF) acting on an object is equal to its mass (m) multiplied by its acceleration (a): ΣF = ma. Since the structure is not moving, its acceleration is equal to zero, and therefore the sum of the forces acting on it must also be equal to zero.

Now, let's apply these concepts to the problem at hand. We can start by labeling the joints and rods in the structure and assigning variables to their lengths, as shown in the diagram below:

http://rsizr.com/etc/structure_diagram.png

Next, we can write out the equilibrium equations for each joint in the structure. For the joints A, B, and C, we can write:

Joint A:
ΣFx = 0 → k1ΔL1 - k2ΔL2 = 0
ΣFy = 0 → -k1ΔL1 + k2ΔL2 + F = 0

Joint B:
ΣFx = 0 → k2ΔL2 - k3ΔL3 = 0
ΣFy = 0 → -k2ΔL2 + k3ΔL3 + F = 0

Joint C:
ΣFx = 0 → k3ΔL3 - k4Δ

 

[Moetasim]

I appreciate your attempt to solve this problem using your own intuition and equations. However, in order to provide a more accurate and reliable solution, we need to use physics principles and equations that are grounded in real-world applications. In this case, the equilibrium lengths of the elastic rods can be determined using the principle of conservation of energy, which states that the total energy of a system remains constant in equilibrium.

To solve this problem, we need to consider the elastic potential energy stored in each rod. This potential energy is given by the equation U = 1/2kx^2, where k is the spring constant and x is the deformation of the rod from its rest length. Since all five rods have the same material and rest length, their spring constants are also equal.

Next, we need to apply the principle of equilibrium, which states that the sum of all forces acting on a system must be equal to zero. In this case, the only forces acting on the rods are the axial forces at the endpoints. Therefore, the sum of these forces must be equal to zero, which gives us the equation F1 + F2 + F3 + F4 + F5 = 0.

Using Hooke's law, F = kx, we can rewrite this equation as kx1 + kx2 + kx3 + kx4 + kx5 = 0.

Since the rods are in equilibrium, the total potential energy of the system must also be equal to zero. This means that the sum of the potential energy stored in each rod must be equal to zero. Using the equation for elastic potential energy, we can write this as 1/2kx1^2 + 1/2kx2^2 + 1/2kx3^2 + 1/2kx4^2 + 1/2kx5^2 = 0.

Simplifying these equations, we get x1 + x2 + x3 + x4 + x5 = 0 and x1^2 + x2^2 + x3^2 + x4^2 + x5^2 = 0.

Using these equations, we can solve for the equilibrium lengths of the elastic rods. However, since there are five unknowns and only two equations, we cannot determine the exact lengths of each rod. We can only determine the ratios between the lengths, which