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jjalexand

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**(I assume that the three section headings below form the template referred to below)**

1. Homework Statement

n identical equi-distant particles are distributed equi-distantly around the circumference of a ring of radius r in space. Each particles is of mass m, so the total mass of the ring is n*m. External gravitational fields are negligible and there is no other mass within the ring. The particles are initially at rest and the ring is not rotating. How long will it take the particles to meet at the center of the ring under gravitational attraction between the particles?

1. Homework Statement

n identical equi-distant particles are distributed equi-distantly around the circumference of a ring of radius r in space. Each particles is of mass m, so the total mass of the ring is n*m. External gravitational fields are negligible and there is no other mass within the ring. The particles are initially at rest and the ring is not rotating. How long will it take the particles to meet at the center of the ring under gravitational attraction between the particles?

## Homework Equations

"Time to meet function" for two particles at rest to meet under the force of gravity:

t = time to meet function =

**pi * (8 * G)^(-1/2) * x^(3/2) * (m1 + m2)^(-1/2)**[/B]

**where**

t is the time to meet

G is the universal gravitational constant

x is the initial distance between the two particles

m1 and m2 are the masses of the two particles

t is the time to meet

G is the universal gravitational constant

x is the initial distance between the two particles

m1 and m2 are the masses of the two particles

**Ref: Answers to Physics Forums question**

## The Attempt at a Solution

radius of ring = r

n particles of mass m

total mass n*m

the centre of mass of the ring is at the physical center of the ring

there are n^2 unique pairs of particles where each pair includes two particles attracting one another

the mutual gravitational attraction of each unique pair is independent of every other pair, i.e. the mutual attractions can be combined linearly and additively

Result for two particles is

**pi * (8 * G)^(-1/2) * (2r)^(3/2) * (m + m)^(-1/2)**Result for three particles is:[/B]

**Side of an equilateral triangle inscribed in a circle of radius 1 is approx 1.732**

There are three identical pairs each 1.732r, but the infall direction is towards the center of the ring not along the straight line between the particles, and it is hard to see how to use the time to meet function in this situation and also to generalize to n particles

There are three identical pairs each 1.732r, but the infall direction is towards the center of the ring not along the straight line between the particles, and it is hard to see how to use the time to meet function in this situation and also to generalize to n particles