# (Mathematical Physics) Mass on four elastic rods

• asynja
In summary, a gravitational detector consists of a 10^4 kg mass placed on four rods in a square formation. The rods, which are 25 cm high with a cross-sectional area of 3cm^2, are made of copper and steel and have polished surfaces at the same height. The test mass is also polished with the same precision. The first question asks for the contraction of the rods when the test mass is carefully placed on them so that its center of mass is directly above the center of the square. The second question inquires about the relationship between the height of the rods and temperature, with specific coefficients of linear thermal expansion given for copper and steel. Finally, the last question asks for the temperature change needed for only two
asynja

## Homework Statement

In a gravitational detector we put a mass of 10^4 kg onto four rods, which are located at the angles of a square. The rods are 25 cm high with S = 3cm^2, two of them are made of copper, two of steele and they are situated on opposite sides. The top surfaces of rods are polished to the exactly the same height, the bottom side of the test mass is polished with same precision.
A) For how much do the rods contract when we carefully place the test mass onto them, so that its center of mass is exactly above the center of the square?
B) How does the height of rods depend on the temperature?
How much does the temperature need to change so that only two rods would carry entire test mass?
E(Cu)=8*10^10 N/m^2, E(steel)=2*10^11 N/m^2, the coefficients of linear thermal expansion are 17*10^(-6) K^(-1) for copper and 12*10^(-6) K^(-1) for steel.

## The Attempt at a Solution

http://img638.imageshack.us/img638/8550/mafinaloga1.jpg
I think my attempt is not correct. I started by supposing that all rods contract to the same lenght.
This is a problem from an old exam (Mathematical Physics) so I don't have the solution. Any help would be appreciated.

Last edited by a moderator:
It would seem the rods must contract the same amount (for the weight not the heat) since otherwise to rods carry all the weight thus the others don't contract thus they do carry weight which is a contradiction.

Ok. How about the rest of my solution? (I'm sorry I had to post a picture, but I'm no good at typing equations and this was easier.)

## 1. What is the concept of "Mass on four elastic rods" in mathematical physics?

The concept of "Mass on four elastic rods" refers to a model in mathematical physics that involves a system of four elastic rods connected to a central mass. This system can be used to study the dynamics and vibrations of physical structures, such as bridges or buildings.

## 2. How is the mass distributed on the four elastic rods?

The mass is evenly distributed along the length of each rod in the "Mass on four elastic rods" model. This means that each rod contributes equally to the overall mass of the system.

## 3. What are the equations of motion for the "Mass on four elastic rods" system?

The equations of motion for this system can be derived using the principles of classical mechanics and elasticity. They involve variables such as mass, length, and stiffness of the rods, as well as the displacement and velocity of the central mass.

## 4. How does the "Mass on four elastic rods" model relate to real-world structures?

The "Mass on four elastic rods" model is often used as a simplified representation of real-world structures, such as bridges or buildings. By studying the dynamics and vibrations of this model, scientists can gain insights into the behavior of these structures and make predictions about their stability and safety.

## 5. Are there any limitations to the "Mass on four elastic rods" model?

Like any mathematical model, the "Mass on four elastic rods" system has its limitations. It assumes a linear relationship between the applied forces and the resulting deformations, which may not always hold true in real-world structures. Additionally, the model does not take into account factors such as damping or non-uniform mass distribution, which may affect the behavior of the system.

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