A classical mechanics problem by Jason Hu
A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume ρ, and occupy a region of radius r. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? The time is expressed as x/sqrt(Gρ), where G is the universal gravitational constant. Find 10000x.
The answer is 5427.
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1 solution
We reduce this collapsing spherical cloud to two particles: one with mass M and another with m'. Mass M represents the mass of the spherical cloud condensed to a "small region". m' represents the "dummy mass" we'll use for convenience( Note: m' is not really important). The problem is essentially how long it will take for mass m' to collide with M?
We can assume that M >> m' so that we can assign an inertial frame at M. Initially m' is a distance, r, from M. Eventually you'll get a semi-tough integral that you'll have to solve. The integral is built on the fact that the instantaneous velocity(v) is equal to v = d t d x . After solving my integral for time, I got t = ( G ρ 3 π / 3 2 )^1/2