Gravitational Motion of Masses on Polygons: n=8 to ∞

[Sakriya]

Suppose a n-sided polygon. Point particles of mass "m" each are placed in the corners of the polygon. How does the system of particles move if the only force anting between them is gravity? After how much time the bodies collide if n= 8 and n tends to infinity?


Any suggestions are welcome .
this is not homework question.

 

[tiny-tim]

Hi Sakriya!

It's symmetric, so they'll obviously move radially inwards.

Can't you just add all the individual forces?

If you don't fancy that, you could instead start by calculating the PE as a function of radius …

what do you get? ​

 

[Sakriya]

Thanks
Adding individual forces would be very long, using PE it becomes easy...
how should i do it when n tends to infinity

 

[tiny-tim]

… how should i do it when n tends to infinity

Can't you just find the formula for n, and let n -> ∞ ?

 

[pmacias]

I would approach this question by first considering the laws of gravitation and motion. According to Newton's law of gravitation, the force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This means that as the number of particles increases (n tends to infinity), the force of attraction between them also increases.

In this scenario, we have n point particles of equal mass placed at the corners of a polygon. Since the only force acting between them is gravity, they will all be pulled towards each other. The direction of motion will depend on the initial placement of the particles, but eventually, they will all collide at the center of the polygon.

The time it takes for the particles to collide will depend on the initial velocity of each particle and the distance between them. If we assume that the initial velocity of each particle is zero, then the time it takes for them to collide can be calculated using the equation t = √(2d/g), where t is the time, d is the distance between the particles, and g is the acceleration due to gravity.

For a regular n-sided polygon, the distance between the particles can be calculated using trigonometry. For n=8, the distance would be d = a/√2, where a is the length of one side of the polygon. As n tends to infinity, the distance between the particles will also decrease, meaning the time it takes for them to collide will also decrease.

In summary, in a system of particles on a polygon where the only force acting between them is gravity, the particles will eventually collide at the center of the polygon. The time it takes for them to collide will depend on the initial velocity and the distance between them, which decreases as n tends to infinity. Further research and calculations may be needed to determine the exact time of collision for n=8 and n tends to infinity.