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A question about volume (caution: Mathematics and physics question)? Answered

 Ok, I know how to figure the "volume of a sphere" and the surface area of said sphere,  now I need to know how to figure how much I need to remove from the outside, to reduce the volume by about 1/2.  

I suspect this will NOT reduce the size of the entire sphere by 1/2....at least, that doesn't seem logical to me.

Can anyone explain it so that I (not a mathematician) can understand it?


 

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CameronSS (author)2010-09-09

Volume is a cubic function, area is a square function, distance is a linear function. If you double the area (linear), the surface area (square) will be quadrupled (22=4) and the volume (cubic) will be octupled, if that's even a word (23=8).
 
To determine linear proportions from volume, work backwards. To make the volume 1/8 of what it was, the diameter must be halved (cube root of 1/8 is 1/2). To make the volume half of what it was, the volume must be the cube root of 1/2 of the original, about 0.79. So if you start with a sphere of diameter 1", a sphere with diameter .79' will have about half the radius.

The same concept is true for all three-dimensional objects, regardless of shapes.

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Goodhart (author)CameronSS2010-09-09

Ok, thanks.....let me see if I can plug that into what I am working on (the numbers are a bit on the large size, on the order of 1,409,147,148,464,966,056.25 CKm, etc. as the volume)....

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CameronSS (author)Goodhart2010-09-09

You know, we already have gravity holding us in place, you don't need to put Gorilla Glue on the sun to win the contest...

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Goodhart (author)CameronSS2010-09-09

Nothing like that.....I have heard someone say that the sun would have to be "double" the size it is now, if it has already used 1/2 of it's fuel, at the very beginning. Now, right away I gave him the ole o_0 look and said I'd get back to him on that (I couldn't do this in my head, and some of it I was unfamiliar with).

Now, I know my figuring will have a "grave error" in it. but it will be to his advantage and I suspect that it will STILL not come even close to what this innumerate, supposed mathematician has come up with. LOGIC tells me he's wrong, without even doing the math.

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CameronSS (author)Goodhart2010-09-09

Yep, flawed logic. He's going off the (incorrect) assumption that as the sun "consumes" fuel, it disappears and does not contribute to the total volume. It's nuclear fusion, which does convert mass into energy, but only a small fraction of the mass. And it would only have to start off with about a 25% larger diameter if his method was correct.

Kelsey probably knows a better answer, but he's slacking off on the forum-answering, apparently. Probably off creating a black hole to eat the solar system or something.

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Goodhart (author)CameronSS2010-09-09

Yes, but even if we assume a reduction of 1/2 over the last, what, 5 X1011 years, we STILL don't get a sun that is doubled in size that would consume (another flawed bit of logic) the planets out to Mars.  

Oooo, in a way I hate pseudo-science, and in a way I like it....it keeps me thinking.... ;-)    I have to work from scratch though....not having a heavily mathematical background....I have to figure it out as I go along. 


So,  I came up with....if 1/2 the volume of mass were removed we'd get a surface area reduction from 6,078,428,661,500 CKM  down to 4,801,958,642,585 CKm   So, even given his false information, the math still doesn't support his determination.

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kelseymh (author)Goodhart2010-09-09

I think you mean 4.6 x 10^9. The whole Universe is only 13.9 x 10^9 years old.

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CameronSS (author)kelseymh2010-09-09

We're talking Universe 2.0.

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Goodhart (author)CameronSS2010-09-11

On the other side of the "great expanse"

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Goodhart (author)kelseymh2010-09-09

Um, yeah, I am still having a few problems with converting to the short hand.... 10 9  = 10,000,000,000  right ?

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CameronSS (author)Goodhart2010-09-09

109=1 000 000 000

The exponent gives you the number of zeros to put after the 1.

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Goodhart (author)CameronSS2010-09-10

Right, that was what I had thought I had done earlier... so 15 billion = 15,000,000,000 = 15 x 109   (don't ask me WHERE I got 11 from....I was kind of tired last night).

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CameronSS (author)Goodhart2010-09-09

Oh, he might be getting confuzzled from the sun's end-of-life process. If I recall correctly, it will eventually start venting matter and expanding due to decreased gravity, and I think the prediction is that when it swells to a red giant, it will head out toward Mars' orbit. Of course, the reduced gravity also means that the orbits will all be farther out, so it gets all weird.

Phil Plait and James Randi would be proud.

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Goodhart (author)CameronSS2010-09-10

Actually, I think I know where he got his "reasoning" from....one moment....nope, I was wrong (or at least, I couldn't find it quickly). I had thought I read something along this line, when much younger in the book: The Creation-Evolution Controversy, by Wysong. But I don't see it there now. I mean, I keep the book, with my own annotations I placed in it, as a reference as to what others "think" (or don't think, as the case may be).

But I can no longer find the reference in any of the books I have remaining in my library. Perhaps it was lost during my last move over 20 years ago.....meaning I don't have need to consult it to demonstrate mistakes much anymore.

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kelseymh (author)Goodhart2010-09-09

It really helps when the actual topic of the question is up in the topic :-/

Caitlinsdad is right, of course. There are so many things wrong with your "someone's" statement, it's almost not worth it to even start.

The Sun does not "consume fuel" like burning wood. The mass difference (binding energy) between four hydrogen nuclei and one helium nucleus is only a few MeV (out of ~4 GeV), so your talking about a 1 part per thousand loss of mass.

The visual size of the Sun is only marginally connected to its mass. Much more important is the Sun's temperature, which determines the pressure that keeps it "inflated" against gravity. Four billion years ago, the Sun was cooler, and is estimated to have been about 1/3 smaller in diameter than it is now.

As for the math, it's pretty simple, and I think it's been explained. The volume of a sphere is just V = (4/3) pi r^3. Since you're dealing with ratios, the (4/3)pi cancels out, and you can write V1/V2 = 1/2 = r1^3 / r2^3, so r1/r2 = 1/cbrt(2) = 0.794. So to reduce the Sun's volume by 50%, you only need to reduce the radius/diameter by 20%.

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Goodhart (author)kelseymh2010-09-09

"It really helps when the actual topic of the question is up in the topic"

Yeah, sorry about that....it's been kind of a hectic month....I am not thinking as clearly as I could be.

Thanks for that.  I got the numbers but hadn't yet converted it to the proper percentage.....and that is the one thing I really needed to know....although, as I wrote to Cameron,  that is about what I'd have guessed....the difference in size is negligible compared to the (incorrect assumption) that it would lose 50 % of it's volume in the first place.

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caitlinsdad (author)CameronSS2010-09-09

Now take a pair of smartypants, like the ones you are wearing. To halve the volume you would think you can just cut off one leg of the pants. But no, there is the waistsize and inseam measurement to figure and will you be wearing a belt or suspenders, none or both. Oh and don't forget if you will be hemming it with cuff or without, straight leg, boot cut or flared...

Thanks for the explanation...but no link to some fancy widget to graphically input some magic numbers and have an animated display to show us the volume of a sphere being halved?

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CameronSS (author)caitlinsdad2010-09-09

Widgets are wimpy.

I want that on a tshirt.

And shaddup. :P

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Goodhart (author)CameronSS2010-09-09

ok, so my figure of solar surface area comes out to about: 6,078,428,661,500 CKm does that compute?

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caitlinsdad (author)CameronSS2010-09-09

I have another t-shirt in mind. :P

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