Pitching Forward

In their house in Ashfield, my parents have a 500-foot well that delivers their water. Growing up drinking that delicious water has spoiled me for most municipal water supplies, including the one here in Gainesville. To appease my picky palate, we got a filter:

The tap water goes in the reservoir at the top, then drains down through the filter to the bottom. It's possible to pour filtered water while some is still in the top area, and it made me wonder how the pitcher's center of mass changes as it tips forward, and as the water drains through.

The first problem is to find the shape of the water when the pitcher is tipped at an angle. The surface of the water will stay horizontal, so if we imagine tipping to an angle θ, we can instead imagine the pitcher flat, with the water at the angle:

Looking from the top, the pitcher is approximately an ellipse:

We're looking for the center of mass of the water, which is given by

where M is the total mass, and ρ is the mass density. This is essentially the weighted average of the position of each bit of water. Since water has uniform density, we can replace ρ and M with the total volume. Figuring out the limits of integration is a bit tricky: We can imagine moving in the x direction in the diagram above (left to right) adding up squares of water. The widths will be

and the heights will be

This makes the integral pretty ugly, so I threw it into Mathematica for an answer:

That's just for one of the water reservoirs; to get the CM for both, we weight the positions by the amount of mass in each. Since we're interested in tipping the pitcher forward, we can measure the angle of the CM from the front edge:

Remember though that we're also tipping forward by θ, so for the tipping point, we're looking for θ + φ > 90°. I put together some Python code to find the CoM angle for various tipping angles and points in the filtering process:

I find it really interesting that under this model, the 90° tipping point stays nearly constant. However, this fails to account for when there is not enough water to cover the bottom of either reservoir. From experience, I certainly don't feel the pitcher falling forward as early as 9 or 10 degrees. I tried to account for that in the integral, but the lack of symmetry makes it too difficult for Mathematica to handle. Even with that missing piece, an interesting system to analyze!