Rolling, Rolling, Rolling
A few days ago I was relating the cans-in-a-blanket problem, and retelling the vacuum joke and story to someone who had not yet heard them. One of my colleagues commented on a problem he had been given during an interview, also involving cans of soda:
You have two cans, one filled with ice and the other with liquid, but otherwise identical. The cans are rolled down an incline. Which one reaches the bottom first?
Much like the previous problem, I think there is a common misconception at play here for some people who get the answer wrong, and I’ll get to the explanation below. One of the people in the conversation said his first impulse was the wrong answer, but when we discussed the physics, we all agreed on the solution.
I set up to do a demonstration, though my first attempt was thwarted — I filled up a can with water and popped it in the freezer, hoping the can would be strong enough to hold together and have the ice expand vertically. It wasn’t.
I think the problem being that since ice will freeze from the top down and outside-in, the ice adhered to the can too well to let it expand upward as much as I hoped. (BTW — Black Cherry Citrus? Blecch. I bought it by accident when they redesigned their color scheme and introduced the flavor)
So I did it again, adding a little bit of water and letting that freeze, repeating the process several times until it was full, and it worked. Here is the experiment to investigate the problem given above:
For those who think that the liquid-filled can will roll more slowly, I think I know what the misconception is: most of us have seen or done the experiment with spinning an egg, and a hard-boiled egg spins readily while the unboiled egg doesn’t. So the intuition is that since liquids don’t spin readily, the liquid-filled can won’t want to roll very fast. And, as we can see, that’s wrong.
The reason the intuition is wrong is from a misinterpretation of the reason the unboiled egg doesn’t spin — it’s because it’s difficult to transfer energy and angular momentum to the liquid by spinning the container; the coupling between them is weak. And angular momentum tells you the tendency for something to spin — it only changes when you apply a torque. With the soda cans it means that the work being done, adding energy (gravity acts on it, and there is a torque from the friction of the treadmill causing rotation)but this energy isn’t being added to the liquid, so it must be going into the can itself, which isn’t very massive — almost all of the energy goes into translational kinetic energy. The frozen water, though, does rotate with the can, so the gravitational potential energy has to be shared between translation and rotation of the can + ice system, so the translational kinetic energy (and therefore speed) is smaller.
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Does the frozen can travel farther than the liquid filled can?
I didn’t test that. It will depend on the frictional losses, and you might expect that the soda in the can would have some additional friction as it interacts with the inside surface, which would bring it to rest in a shorter distance.
Ice is less dense, so it weights less?
The soda can isn’t quite full — the masses were very close. That’s not the reason for the speed difference.
The reason a raw egg won’t spin is because the liquid yolk is affixed to the shell by two bands of chalazae, which causes a dampening effect.
The liquid stays to outside wall of the can and when it rotates to the top of the can it has a greater mass that is being pulled by gravity as the can spins and the inside liquid is pulled down. The frozen can has uniform mass so gravity as evenly.
Damn I guessed wrong. I see now that the ice can rolls slower because the ice has to spin up. This requires energy which reduces the speed. The liquid does not spin up as much, so less energy is required, hence more energy for speed.
The can with liquid initially has a lower moment of inertia (because, as you said, the mass of the liquid isn’t tied to the can itself), so the rolling can start much sooner than the can with a solid. This test, however, fails to prove in any way which can will win over a long distance. As the speed of rotation increases, the factors involved in the speed become much more complex. I’m predicting that over any significant distance, the moment of inertia of the can with the liquid quickly becomes larger than that of the solid due to internal friction.
My intuition lead me astray as well; I was thinking about damping in the liquid. It should be interesting to test rolling on a flat surface.