Volume of a Partial Sphere
Date: 01/07/98 at 18:50:46 From: Thomas Lee Subject: Volume of a partial sphere If I take a spherical container filled with fluid and then introduce a bubble of gas into the container (which displaces the fluid so the bubble's volume remains unchanged), what would be the surface of the sphere which is in contact with the bubble (assuming the bottom meniscus of the bubble is flat - i.e. that part which is not in contact with the container) given the volume/radius of the sphere and the volume of the bubble? I am asking this because I will be giving a talk on the use of intra-ocular gases in eye surgery.
Date: 01/08/98 at 08:35:32 From: Doctor Jerry Subject: Re: volume of a partial sphere redux Hi Thomas Lee, As I understand the problem, you have a sphere of radius a (this is the container) into which is put a (small) bubble of gas, say of radius b. In my visualization, this small bubble rises to the top of the container and deforms to something fitting against the sphere on top and having a flat bottom. I take it that the physics of the situation makes this at least approximately true. You ask "what would be the surface of the sphere which is in contact with the bubble." My eye was drawn to the word "surface," hoping that you meant surface and not volume. Of course, either could be calculated. So that I can make my terminology clear, imagine a circle of radius a (a cross-section of the container). Draw a horizontal line ABOVE the center of this circle. Suppose the horizontal line is h units from the top. I stress above since formulas may change if the horizontal line is below the center. I assumed above since the bubble is probably relatively small. Let me know if this isn't the case. The chunk of the container lying above the plane corresponding to the line is called a spherical segment. Its volume is V=(pi*h^2/3)(3a-h) and its (top) surface area is S = 2*pi*a*h. You said that the bubble doesn't change volume as it deforms. Okay, so whatever its shape, its volume is (4/3)*pi*b^3. So, (4/3)*pi*b^3 = (pi*h^2/3)(3a-h), where a and b are known. We can solve for h (it's just a quadratic) and, then, knowing h we can calculate S = 2*pi*a*h. I'd be grateful if you would send a follow-up message. For one thing, I'd like to see if you agree with my calculations and my assumption about the size of the bubble. I'd also appreciate more detail about the circumstances of this calculation. I'm asking since I'm co-authoring a calculus book and would like to use this example as a project in the book. So, if I can attach some genuine application words and background, all the better. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.