How to Solve an Inverse Square Law Problem in Physics?

AI Thread Summary
The discussion focuses on solving a physics problem involving a particle attracted by an inverse-square-law force in one dimension, modeled after a linear particle accelerator. The force acting on the moving object 'O' is defined as F = S / (d - x)², where 'S' represents field strength, 'd' is the position of the fixed source 'M', and 'x' is the position of the projectile. The user seeks guidance on finding a closed-form solution for the object's position over time, suggesting methods like calculating potential energy, deriving the Lagrangian, and applying the Euler-Lagrange equation. They also mention an alternative approach using a second-order differential equation derived from Newton's second law. The discussion emphasizes the need for direction rather than a complete solution.
antiduh
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Howdy folks, I'm trying to solve a physics problem that I posed to myself one day after a class. My inspiration is a linear particle accelerator.

So, what I want to do is model a particle in one dimension, being attracted by an inverse-square-law force.

Code:
|<---------d-------->|
|<--x-->|            |
|       |            |
|-------O------------M
|       ^            ^
        |            |
Projectile           |
                     |
Attracting point source
Object 'M' is the attracting point source, such as a magnet. Object is fixed.
Object 'O' is the object being attracted, obviously free to move.

Distance 'd' is the position of M.
Distance 'x' is the posiition of the projectile.

So, let us assume that the force on 'O' is:
Code:
       S
F = -------
    (d-x)^2
Value S is a unitless abstraction of field strength.


The problem is thus: Find a closed-form solution for the position of the object at time 't'.


Any help would be appreciated. I'd prefer someone to just point me in the right direction, and not solve the whole thing for me, but I'll take whatever help I can get.
 
Last edited:
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i have lost touch with how to do this but i think you do this

find the potential first V = \int F \cdot dl

then find the Lagrangian L = T - V

Use Euler Lagrange equation to solve for this trajectory

but i could be wrong...
 
an easier way is

F = m \frac{d^2 x}{dt^2} = \frac{S}{(d-x)^2}

you now have a linear second order DE which you need to solve
 
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