What is the Hooke's Law Equation for a 3D System with Attached Spheres?

In summary, the conversation discusses the use of Hooke's law in a 3D system with attached spheres. The question is whether the equation should use the displacement of the sphere of interest from its neighboring sphere, or vice versa. The speaker notes that this becomes complicated when considering multiple spheres, and it can lead to messy and unsymmetrical equations. They mention the classic three body problem as an example of simplicity in a complex system, and express interest in seeing different perspectives on the issue.
  • #1
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I am working on a theory and this thing is bothering since the past few hours...

When we write down hooke's law that is

d{2}x/dt{2} = -kx

We write down as x as the displacement from the mean position given that the mean position coincides with zero...

Now let's suppose that i have a 3D system.. And a sphere attached with various spheres... Let's say that the sphere we are interested in is (i,j). And out of the several spheres attached to ij let's say we take (i,j-1).

The question is what is my hookes law equation.. Is it

1) d{2}r{ij}/dt{2} = -[some constant][r{i,j} - r{i,j-1}]

or

2) d{2}r{ij}/dt{2} = -[some constant][r{i,j-1} - r{i,j}]

Now it is not that easy.. Cause i need to generalize this.. I have several other spheres attached with (i,j).. And if i use 1 it causes problem with some of the spheres and is okay for the rest... And if i use 2 it causes the same problem...

I can't use 1 for some spheres and 2 for others... Cause then in my 3D infinite space the equations will depend on the position of the spheres...And the equations would be a mess to solve and the most important it loses its beauty...
 
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  • #2
When things get complicated the mathematical beauty and symmetry has a tendency to fade. In some cases though, new beauty and symmetry can emerge out of complexity. In this case I can't help but think of the classic 3 body problem. I think you can get some simple and describable modes out of such a system, but in general - yea, it gets messy.

Im interested to see some real replies. :p
 
  • #3
Academic said:
When things get complicated the mathematical beauty and symmetry has a tendency to fade. In some cases though, new beauty and symmetry can emerge out of complexity. In this case I can't help but think of the classic 3 body problem. I think you can get some simple and describable modes out of such a system, but in general - yea, it gets messy.

Im interested to see some real replies. :p

The three body problem is elegant... But in that case there are no complexities involved cause there are just three equations and its easy solving them... So looking at the three body problem wouldn't help...
 

1. What is Hooke's Law?

Hooke's Law is a physical law that states the force required to extend or compress a spring is directly proportional to the distance the spring is stretched or compressed.

2. Who developed Hooke's Law?

Hooke's Law was developed by English scientist Robert Hooke in the 17th century.

3. What is the equation for Hooke's Law?

The equation for Hooke's Law is F = -kx, where F is the force applied, k is the spring constant, and x is the distance the spring is stretched or compressed.

4. What is the significance of Hooke's Law?

Hooke's Law is significant because it helps us understand and predict the behavior of springs and other elastic materials. It also serves as a fundamental principle in the field of mechanics and is used in many engineering and design applications.

5. Are there any limitations to Hooke's Law?

Yes, there are limitations to Hooke's Law. It is only applicable to materials that behave elastically, meaning they return to their original shape after the force is removed. It also assumes that the force and displacement are directly proportional, which may not always be the case in real-world scenarios.

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