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#10 seems to have confused a number of people, so I'll share my thoughts. The reason that decreasing the amount of liquid lowers the center of gravity is that

#10 seems to have confused a number of people, so I'll share my thoughts. The reason that decreasing the amount of liquid lowers the center of gravity is that you're removing liquid from above the center of gravity. Likewise, the center of gravity rises again when you remove liquid from below the center of gravity. Thus, the center of gravity is lowest when the center of gravity is exactly at the level of the liquid. Without the distribution of the weight of the can, I don't think we can get an actual number. The ratio of the weights of the top/bottom to the weight of the sides is important. A final unrelated point: cans of soda are measured in fluid ounces, so the proposed numbers are somewhat less accurate than the questioner seems to think.

Strilanc 12y ago
> Without the distribution of the weight of the can, I don't think we can get an actual number. Because the distribution of weight in the can isn't changing, you can treat it as a point mass as far as the center of mass is concerned. The fluid's distribution is changing, but we know how.
- thedufer 12y ago
  Hrmmm. I did a full write-up assuming that worked, and then managed to convince myself it didn't. Looks like you're right, though. Here it is: Let's call C the weight of the can, L the weight of the liquid, and x the proportion of the can that's full. Then the center of mass is (C/2 + xL/2)/(C+L). We want where this is equal to x. Solving for x gives us x=C/(2C+L). This passes the sniff test: for C>>L, x->1/2, and for L>>C, x->0, as we'd expect. For the given weights, we get x=0.1, so when the can is 10% full.
  - Strilanc 12y ago
    The liquid's center of mass isn't xL/2, it's x L^2 /2. One factor of L for getting lighter, and one for getting lower. I made the same mistake like five times.
    - thedufer 12y ago
      In my formulation, x is the proportion and L the total weight, so I believe it would be x^2 L/2. I also just realized that the denominator ought to contain xL, not L. My solution is a mess, clearly. The original idea is right, though.
jameshart 12y ago
The point of a fluid ounce is that it's the amount of water which weighs an ounce - roughly (allowing for various failures of standardization and definitions of water purity and measurement circumstances). And most beverages have a specific gravity close enough to 1.0 that you can consider them more or less water. So yes, the weight of the liquid in a 12oz beverage is, for puzzle purposes, pretty much 12oz.
- thedufer 12y ago
  That was the point hundreds of years ago, but its now fairly inaccurate. For the purposes of food labeling, a fluid ounce is defined as 30mL, which makes a fluid ounce of water weigh almost 5% more than an ounce. Sodas are typically more dense - Coke is 10% more dense than water, for example, for a total weight of over 13.5 ounces for a 12 fl oz can. Seems like a pretty big difference to me.

Strilanc 12y ago

> Without the distribution of the weight of the can, I don't think we can get an actual number. Because the distribution of weight in the can isn't changing, you can treat it as a point mass as far as the center of mass is concerned. The fluid's distribution is changing, but we know how.

thedufer 12y ago
Hrmmm. I did a full write-up assuming that worked, and then managed to convince myself it didn't. Looks like you're right, though. Here it is: Let's call C the weight of the can, L the weight of the liquid, and x the proportion of the can that's full. Then the center of mass is (C/2 + xL/2)/(C+L). We want where this is equal to x. Solving for x gives us x=C/(2C+L). This passes the sniff test: for C>>L, x->1/2, and for L>>C, x->0, as we'd expect. For the given weights, we get x=0.1, so when the can is 10% full.
- Strilanc 12y ago
  The liquid's center of mass isn't xL/2, it's x L^2 /2. One factor of L for getting lighter, and one for getting lower. I made the same mistake like five times.
  - thedufer 12y ago
    In my formulation, x is the proportion and L the total weight, so I believe it would be x^2 L/2. I also just realized that the denominator ought to contain xL, not L. My solution is a mess, clearly. The original idea is right, though.

jameshart 12y ago

The point of a fluid ounce is that it's the amount of water which weighs an ounce - roughly (allowing for various failures of standardization and definitions of water purity and measurement circumstances). And most beverages have a specific gravity close enough to 1.0 that you can consider them more or less water. So yes, the weight of the liquid in a 12oz beverage is, for puzzle purposes, pretty much 12oz.

thedufer 12y ago
That was the point hundreds of years ago, but its now fairly inaccurate. For the purposes of food labeling, a fluid ounce is defined as 30mL, which makes a fluid ounce of water weigh almost 5% more than an ounce. Sodas are typically more dense - Coke is 10% more dense than water, for example, for a total weight of over 13.5 ounces for a 12 fl oz can. Seems like a pretty big difference to me.