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 -> ∞ ?