Graviational potential of 2 cones

Homework Statement

Consider two hollow fixed cones (such as ice cream cones without the ice cream). They have a base radius R, slant height L,and a surface mass density σ. The cones are vertical, with their apexs touching (say, at the origin). A particle initially at rest falls in from infinity, along a perpendicular bisector line. What is its speed when it reaches the tip of the cones?

F = GMm/r^2
V= GMm/r

The Attempt at a Solution

So I am trying to write an equation for the cone, where if I pick any arbitrary height on the cone, I would get the circumference at that point. I would then integrate over the height of the cone. However, I am having trouble coming up with such an equations. I know that the forces in the z and y direction will cancel, so the particle will be "pulled" towards the vertex of both cones.

Would it be possible to consider the cones as a point at their center of mass?