[math-fun] A problem for the shower
A question struck me in the shower this morning: Suppose you would like a piece of soap which dissolves at a uniform rate. I.e., as it gets smaller, the soap bar's surface area remains constant until it vanishes. I have convinced myself that one should be able to construct such a bar of soap with arbitrarily small holes. As the soap gets smaller, the holes are exposed to increase the surface area. Need it have holes?
People in the solid rocket business discovered this by accident (literally!). If the burning discovers a crack, the rate of burning goes way up & the rocket blows up. You might check the geometry that these rockets use -- I would imagine that most would want to keep the overall pressure relatively constant, so the burning surface area should remain constant, as well. At 09:26 AM 4/26/2006, David Wolfe wrote:
A question struck me in the shower this morning:
Suppose you would like a piece of soap which dissolves at a uniform rate. I.e., as it gets smaller, the soap bar's surface area remains constant until it vanishes.
I have convinced myself that one should be able to construct such a bar of soap with arbitrarily small holes. As the soap gets smaller, the holes are exposed to increase the surface area.
Need it have holes?
On 4/26/06, Henry Baker <hbaker1@pipeline.com> wrote:
People in the solid rocket business discovered this by accident (literally!).
If the burning discovers a crack, the rate of burning goes way up & the rocket blows up.
You might check the geometry that these rockets use -- I would imagine that most would want to keep the overall pressure relatively constant, so the burning surface area should remain constant, as well.
The typical way to do that with a solid rocket motor is to begin with a star-shaped cross section, which burns to a circular section at the casing. The perimeter of the star is the same as the circumference of the circle, so there is even burning (and constant thrust) throughout the burn of the motor. Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry
Consider a hollow cylinder. The outside surface loses area faster than the inside because dA/dr=kr. Now twist it to make a Mobius strip: there are no longer two different surfaces. The only problem left is its finite width, which I don't see how to get around. Besides, a wide, thin Mobius strip would make a weird cake of soap. How about a Klein bottle? Would a four-dimensional person need a shower? Steve Gray ----- Original Message ----- From: "David Wolfe" <wolfe@gustavus.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, April 26, 2006 9:26 AM Subject: [math-fun] A problem for the shower
A question struck me in the shower this morning:
Suppose you would like a piece of soap which dissolves at a uniform rate. I.e., as it gets smaller, the soap bar's surface area remains constant until it vanishes.
I have convinced myself that one should be able to construct such a bar of soap with arbitrarily small holes. As the soap gets smaller, the holes are exposed to increase the surface area.
Need it have holes?
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participants (4)
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David Wolfe -
Henry Baker -
Kerry Mitchell -
Steve Gray