# Inverse Square Law problem

• antiduh
In summary, the conversation is about a physics problem involving a linear particle accelerator and a particle being attracted by an inverse-square-law force. The problem is to find a closed-form solution for the position of the particle at time 't'. There are two suggested methods, one involving finding the potential and using Euler Lagrange equation, and the other involving solving a linear second order differential equation. The speaker is seeking help with the problem.

#### antiduh

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:
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

## 1. What is the Inverse Square Law problem?

The Inverse Square Law problem is a mathematical concept that describes the relationship between the intensity of a physical quantity, such as light or gravity, and the distance from the source of that quantity. It states that the intensity of the quantity is inversely proportional to the square of the distance from the source.

## 2. How does the Inverse Square Law affect light?

The Inverse Square Law states that as the distance from a light source increases, the intensity of the light decreases in proportion to the square of the distance. This means that the farther you are from a light source, the dimmer the light will appear.

## 3. How is the Inverse Square Law used in physics?

The Inverse Square Law is a fundamental principle in physics that is used to describe the behavior of many physical phenomena, including light, sound, and gravity. It is used to calculate the intensity of these quantities at different distances from their sources.

## 4. What are some real-world examples of the Inverse Square Law?

The Inverse Square Law can be observed in many real-world situations. For example, the brightness of a light bulb will appear to decrease as you move farther away from it. Similarly, the sound of a loudspeaker will become quieter as you move away from it. The gravitational pull of a planet also follows the Inverse Square Law, as the force between two objects is inversely proportional to the square of the distance between them.

## 5. How do scientists use the Inverse Square Law to solve problems?

Scientists use the Inverse Square Law to calculate the intensity of physical quantities at different distances from their sources. This can be applied in a variety of fields, such as astronomy, acoustics, and radiation physics. By understanding the relationship between intensity and distance, scientists can make predictions and solve problems related to these physical phenomena.