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

Click For Summary
SUMMARY

The discussion focuses on the application of Hooke's Law in a three-dimensional system involving spheres. The user presents two potential equations for the acceleration of a sphere (i,j) in relation to an adjacent sphere (i,j-1), questioning whether to use d{2}r{ij}/dt{2} = -[some constant][r{i,j} - r{i,j-1}] or d{2}r{ij}/dt{2} = -[some constant][r{i,j-1} - r{i,j}]. The complexity arises from the need for a generalized equation that maintains mathematical beauty and symmetry across multiple spheres. The user draws parallels to the classic three-body problem, noting that while it is elegant, it lacks the complexities of their current scenario.

PREREQUISITES
  • Understanding of Hooke's Law and its mathematical formulation
  • Familiarity with differential equations and their applications in physics
  • Knowledge of three-dimensional coordinate systems and vector mathematics
  • Conceptual grasp of the three-body problem in classical mechanics
NEXT STEPS
  • Research the generalization of Hooke's Law for multi-body systems
  • Explore advanced topics in differential equations relevant to mechanical systems
  • Study the mathematical frameworks used in the three-body problem
  • Investigate computational methods for simulating complex mechanical systems
USEFUL FOR

Physicists, mathematicians, and engineers interested in the dynamics of multi-body systems and the application of Hooke's Law in complex scenarios.

FedEx
Messages
318
Reaction score
0
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...
 
Physics news on Phys.org
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
 
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...
 

Similar threads

Graduate Hooke's law, Bertrand's theorem and closed orbits
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Is Hooke's law really not working, or is it me being dummy?
  • · Replies 13 ·
Replies
13
Views
2K
High School Why Does Hooke's Law Use a Negative Sign in Scalar Form?
  • · Replies 30 ·
2
Replies
30
Views
3K
Undergrad Sound pressure and inverse distance law
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Static sphere with gravitating fluid
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Formula for the energy of elastic deformation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad The Feynman way of explaining Symmetry in Physical laws
  • · Replies 45 ·
2
Replies
45
Views
4K
Insights How to Apply Newton’s Second Law to Variable Mass Systems
  • · Replies 0 ·
Replies
0
Views
2K
Undergrad Pressurized containers: Stress distribution and large displacements
  • · Replies 69 ·
3
Replies
69
Views
5K
Undergrad Gauss' law seems to imply instantaneous electric field propagation
  • · Replies 29 ·
Replies
29
Views
2K