Gravitational force due to sphere

[CyberShot]

Homework Statement

Consider a thin hollow fixed spherical shell of radius R and surface mass density rho. A particle initially at rest falls in from infinity. What is its speed when it reaches the center of the shell?

(Assume that a tiny hole has been cut in the shell to let the particle thru.)

Known:

rho, distance between centers of masses, R, G

Homework Equations

F = -GMm/r^2

The Attempt at a Solution

The source of the force acts through the center of the sphere, right? So I know the force, but the problem says the particle falls from infinity (I'm assuming I'd need to use calculus here then)

The gravitational force from the sphere acting on the particle is originating from the sphere's center of mass, and pulling it towards that center radially. The part that tricks me is bringing the test particle in from infinity, so the g-force is pretty much zero at that point in time, and slowly increases at it gets closer. Seems like I require an infinite sum here, but not sure how to realize that.

 

[ehild]

Use conservation of energy. At infinity, the gravitational potential energy is zero, and the particle is in rest, so its KE=0, too. What is the formula for the gravitational potential energy?

When reaching the sphere, and entering into it, getting at a distance r<R from the centre, only that mass attracts the particle that is confined in the sphere of radius r. What is the force in the empty sphere?

ehild