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Topic: Cone floating in water (Read 4296 times) |
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jabhiji
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Cone floating in water
« on: Jan 30th, 2003, 7:51am » |
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A cone shaped wooden object (uniform density) is immersed into a bucket of water. Can the cone float with its apex upwards / downwards / neither / both ?
Does the answer change depending upon the aspect ratio of the cone ?
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Phil
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Re: Cone floating in water
« Reply #1 on: Jan 30th, 2003, 8:24am » |
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A cone with a radius of 10 inches and a height of 1 inch will clearly float either pointing up or down. A cone with a radius of 1 inch and a height of 10 inches will clearly float on it's side. I would guess that any cone that can be floated pointing up can also be floated pointing down. But an exact answer depends on what point the center of gravity is far enough from the base so that small movements of the water allow the c of g to move out from under or over the base and flip onto its side. And that depends on the stillness of the water.
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aero_guy
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Re: Cone floating in water
« Reply #2 on: Jan 30th, 2003, 8:27am » |
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hmmmm, it seems to me that the aspect ratio is important, but so is the density of the wood relative to water. Taking the idea that the object will float in whatever orientation gives it's center of mass the lowest position we may look at the extreme cases. First, almost zero density wood will have it resting on top of the water.
The CG is about 20% from the base, so it is either going to rest on its side or it's base, (obviously not its point). This depends on whether the distance to the edge from the CG is less than the distance to the base. With a little hand waving we see that (if I got the math right) the critical h/r is around 76%. Higher than that it sits on its side, less on its base (of course either would be a stable solution for this case but we are talking which is IDEAL).
OK now, lets look at the case where it is just slightly less dense than water and almost all the material is submerged so we get exactly the reverse situation, with the same critical ratio. So, without getting too into the math we can say it depends on both the relative densities and aspect ratio and a combination can be found to yield any orientation.
QED, or close enough for me. Sheesh I am getting sloppy, no rigorous math this time or anything.
Whoops, Phili didn't see yours, that is what you get for writing out an equation while writing the response.
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| « Last Edit: Jan 30th, 2003, 8:28am by aero_guy » |
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jabhiji
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Re: Cone floating in water
« Reply #3 on: Jan 30th, 2003, 9:11am » |
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I agree with the conclusions drawn for the extreme cases. The math for figuring stability/instability for the intermediate cases seems a bit intimidating.
I wonder whether stability in some orientations of the cone would be enhanced if the cone constantly spins about its axis. (neglecting fluid friction)
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Phil
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Re: Cone floating in water
« Reply #4 on: Jan 30th, 2003, 10:25am » |
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I obviously didn't put much thought into my answer. It might still be correct when the density is about half of water, but when the density is much lower than that, a cone will not float pointing down (imagine trying to float a styrofoam cone pointing down, even if it has a broad base; but if the base is broad enough, and the density isn't quite that low, it will only tilt a few degrees), so, conversely, when density is close to water, it will not float pointing up. A quick search tells me that "The density of water (at 39.2 deg F) is 62.4 lbs/cu. ft., whereas the density for North American wood varies from 17.5 for corkbark fir to 64.9 lbs/cu. ft. for black ironwood." Typical woods seem to be more in the 50-80% of water range.
A spinning cone would have some gyroscopic resistance to tipping, but I can't quite imagine the effect of spinning a top that's floating in frictionless liquid.
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| « Last Edit: Jan 30th, 2003, 10:31am by Phil » |
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jabhiji
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Re: Cone floating in water
« Reply #5 on: Jan 30th, 2003, 11:47am » |
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Interesting thoughts, Phil. I can think of a simple experiment to test your hypothesis. Take a thickish piece of paper and roll it into a conical shape. Once the cone is done, it can be filled with water. Keeping this inside the freezer will provide a cone of ice.....It can then be checked which way it floats.
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aero_guy
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Re: Cone floating in water
« Reply #6 on: Jan 31st, 2003, 12:02pm » |
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By my figuring the only benefit of spining would be to make an unstable equilibrium, either point straight up or point straight down, stable. It would would work in no other orientation other than an axis of symmetry as it would be displacing liquid as it moved and thereby energy and eventually stop spinning even without friction. Hmm, maybe if the liquid were massless.... but then we don't have a buouyancy effect anymore.
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