Beer drinking: inspired by Martin Gardner

I have a can of beer, the can has a mass of \(12\)g and it holds \(495\)ml of beer.

Because I'm on a picnic, and the ground is a bit uneven, I don't want to put down a near-full can, it will fall over and spill! It occurs to me that when the can is 100% full the centre of mass is at the centre of the can. As the beer is drunk the centre of mass will move downwards as the concentration of mass (the beer) gets lower. However, when the can is totally empty the centre of mass will be back at the centre of the can. There must be some point where the centre of mass it at its lowest point and this will be the point where the can is most stable.

If I want to ensure the centre of mass is as low as possible, thus making the can least likely to fall over, how much beer should I drink?


Give the answer in ml.

Assume the density of the beer is constant and equal to \(1\)g/ml.

Also, overlook the fact that the mass of the can, not just the location of the centre of mass, affects stability. For example, in real life a full can might be most stable because it is heaviest and thus harder to push over. The question is really just about finding the lowest point of the centre of mass as the beer is drunk.


Many people will recognise that this is a problem straight from Martin Gardner, however Gardner's solution is actually incorrect, and he specifies many more things in the problem that confuse the working.


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