- #1

- 44

- 0

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}*∑

_{i}ρV

_{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