Full disclosure: I’m no expert on anything, so someone with actual knowledge please correct me. You can think of the egg white as a viscous fluid. The shell is suspended in the fluid. When you push your finger in to fetch the egg, you push the fluid out of the way with your finger. Because if the high viscosity, the fluid moving out of the way of your finger brings the egg shell piece along with it.
Because of how small the piece is. And of the thickness of the egg white (or egg stuff, as you put it).
So you’re trying to pick up this tiny little thing, which already on a dry table top can be hard just because it’s small and difficult to hold.
Well now when it’s in the egg white. When you reach into the egg white, you push some of the egg white out of the way to make room for your finger, when you do this because the egg white is kind of thick, compared to water, the egg white you’re pushing also pushes the egg white around it, which then pushes the on the little piece of egg shell.
So while in water the way will more easily from around your finger and the shell, the egg white is getting pushed forward by your finger, which then pushes the egg steel piece forward too.
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