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## Homework Statement

To show how conservation of angular momentum applied to to a collapsing MC core implies flattening during cloud collapse consider the equation of motion of a parcel mass m in a rotating initially spherical cloud of total mass M, initial angular speed w

_{0}and radius r

_{0}. Consider the acceleration experienced by a particle of mass (m) on the equator of a spherical cloud spinning about a vertical axis relative to that at the poles. You should be able to show that the inward acceleration is less at the equator than at the poles and hence the cloud will inevitably flatten in the equatorial plane during Jeans collapse.

## Homework Equations

Show that the collapse of the cloud will stop in the plane perpendicular to its axis of rotation when the radius reaches r

_{f}= (w

^{2}

_{0}r

^{4}

_{0})/GM

_{r}where M is the mass and w

_{0}and r

_{0}are the original angular velocity and radius of the surface of the cloud. Do this by balancing gravitational and centripetal forces so that a parcel of mass at the equator becomes weightless.

{Hints: (1) the equation of motion in this case is m(d

^{2}r/dt

^{2}) = -((GM

_{r}m)/r

^{2}) + mrcos(theta)w

^{2}; (2) angular momentum is conserved during collapse so that I

_{0}w

_{0}= Iw, where I is the moment of inertia for a uniform sphere of mass M and radius r and theta is the colatitude = 90˚-latitude}

## The Attempt at a Solution

So first of all, I'm not particularly sure what this is asking me to do. I'm trying to use the equations for gravitation and centripetal force to achieve the equation for the radius of when collapse stops:

Since the mass is at the equator, theta = 0, so cos(theta) = 1, therefore I get:

GM

_{r}m/r

^{2}= mrw

^{2}

But I have no idea what to do from here... any thoughts? Thanks! Sorry for the long problem.