A Hill of Beans

I missed posting last week, since the idea I had turned out to be more complicated than I thought, and I ran out of time. I thought I'd be able to finish it this week, but it's still giving me trouble. Luckily, my niece Vivian came to my rescue with a great question (as dictated to my big sister, Rachel):

Vivian has a question about refried beans. She helped me open the can and then tried to dump the beans into the pot. Of course nothing happened. She was delighted and confused when I punctuated a hole in the bottom of the can and the beans slide out in a perfect can shape and landed in the pot with a satisfying plop. What happened?

They sent along a picture to illustrate (and give Vivian an opportunity to mug for the camera):

What we're interested in is the forces on the beans before and after the hole is punched. We can draw the ubiquitous intro-Physics free body diagram:

We have the force of gravity pulling down, and air pressure pushing up. The air pressure, P, is acting over the area of the can's opening, A. Since the beans don't come out of the can, the air pressure must be bigger:

If we express the mass of the beans as a density, 𝜌, multiplied by the volume, which is the area times the height, we come up with

I measured a normal-sized can to be about 11 cm tall, so plugging that in with 1 atmosphere of pressure gives a density of about 94 g/cm^3. For comparison, we can look up some typical densities: water has a density of 1 g/cm^3, and even solid gold is only 19 g/cm^3! If the bar is as high as it is, how can we ever get something out of a can?

That brings us to the second part of the question: What difference does the hole make? The key is that the pressure is only acting on the outside – If any air gets into the can, the pressure will push from the top too, and cancel out the force from the bottom. Punching the hole lets enough air in to allow the beans to slide out. The only reason you even need the hole is that refried beans are both sticky and wet (which sounds like some great ad-copy, right?), so they stay in one piece, and don't allow air to enter around them. The same type of thing would happen with anything that stays can-shaped, like cranberry sauce, or pumpkin puree. The moment any air gets to the back of the can though, the forces cancel, and stuff slides out.

Thanks for a great question, Vivian! I hope you stay curious about the world around you – There's so much to learn!