Center of Mass for a Bowling Ball

[Yoni V]

Hey everyone!
I recently worked through finding the moment of inertia of a cylinder with a cylindrical cavity off center, and I wish to further deepen my understanding of calculating the center of mass of such objects. So I made up the following problem:

Homework Statement

Find the center of mass for a bowling ball of radius R and uniform density, with 3 similar holes characterized as follows:
a. Spherical of radius r.
b. Situated such that their centers are equidistant with distance d (>2r).
c. This is a little tricky for me to explain, because English is not my mother tongue: I want the holes to lie partially outside the ball (like a real bowling ball). Slightly more rigorously speaking, if the ball were a circle, the holes would be circles that intersect the "bowling circle" with right angles.
I hope this description is satisfactory.

Homework Equations

R_cm=1/ρV_total*∑_iρV_i

The Attempt at a Solution

Inspired by the cylinder problem, I tried to treat this as a full ball minus the cavities, but there are few key differences to consider that make it a whole lot more difficult:
a. There is more than one cavity.
b. The cavities are aligned with a spherical symmetry rather than a cartesian symmetry, which I find harder to make sense of.
c. There is considerable complexity in the fact that the holes are partial spheres. I could approximate the holes to be half spheres, though I'd rather find the exact solution.

A similar approach would be to add another "phantom" spherical shell with zero mass around the ball, tangent to the continuation of the holes, and then subtract the holes entirely, although then we would need again to subtract the partial phantom leftover, which gets us back to the start.

Another line of thought is to reduce the problem to simpler forms and then build up the real thing:
a. A bowling ball with one hole instead of three, to isolate the partial sphere mess from symmetry issues.
b. Reducing it to 2d.

But even then, I'm still stuck on the partial sphere/circle crux.

Thanks!
Yoni

 

[mfb]

d>2r? I guess that should be d<2r. And if you don't measure along the curved surface, you even need $d<\sqrt{3}$ (and much smaller for a realistic bowling ball).

Inspired by the cylinder problem, I tried to treat this as a full ball minus the cavities

That is certainly a good start.

a. There is more than one cavity.

If you can calculate one, the other two are not hard.

b. The cavities are aligned with a spherical symmetry rather than a cartesian symmetry, which I find harder to make sense of.

You'll need their position relative to some reference frame (like one centered at the center between the three holes)

c. There is considerable complexity in the fact that the holes are partial spheres.

This effect should be very small for realistic holes. To a first approximation the holes are cylinders. A better approximation considers some tilted end of those cylinders. And if you want to consider deviations from this due to curvature, you first have to find out if the mass distribution and the spherical shape are known well enough to reach this precision at all.

I suggest to start with the cylinders (with some "average" end for the sides where they are open).