Help: Paraboloid volume calcs

Owen Lowe wrote: (clip) the area to solve for is in white with the "?". ^^^^^^^^^^^^ Owen, I think this is a fairly simple calculus problem, but I don't think the information you give is enough to determine a unique solution. A parabola can have a wide, flat contour, like a watch-glass, or a narrow tall shape like a liqueur glass. The solution will depend on the equation describing the shape. The usual form of a parabolic equation is y=4ax. By changing "a" you determine how fast the parabola gets wide. Also, while you refer to the "area to solve for," I am sure you really meant to say "volume." The only reason I mention this is that you could calculate the area of the white part also, but that would not hold ashes.

Hi Leo. yep shoulda said volume, not area.

I agree about the variances of shape and don't really have an answer... What I'm really looking for is a ballpark figure so I can give the hollowing a go and then test with a cup of rice or somesuch - I just want to make sure I have enough depth to the blank as it's going to be a segmented piece.

The Thomas J . Glover Pocket Reference booklet gives the volume of a paraboloid as:

(pi/8) x d squared x h

_____ American Association of Woodturners Cascade Woodturners Assoc., Portland, Oregon Northwest Woodturners, Tigard, Oregon _____

"Owen Lowe" wrote : (clip) I agree about the variances of shape and don't really have an answer... ^^^^^^^^^^^^^^ Owen, I have thought about it some more, and I was hoping to get back before someone noticed my stupid answer. I'm going to check your formula, but I know what I said is wrong, and I'll be back later.

Leo Lichtman wrote: (clip) I'm going to check your formula, but I know what I said is wrong, and I'll be back later. ^^^^^^^^^^^^ I'm back. It's two in the morning, but I think I have it now. Your formula is right. Your answer does not agree exactly with mine--I got 4.09 inches for the height. I used 1 gal = 231 cu in, so dividing by 16, 1 cup = 14.44 cu in. However, I don't think you will probably turn an exact paraboloid on the lathe, so I suggest going a little over. The worst thing would be to pour the ashes in, and have some left over. A little free space in the top of the cup surely would be desirable.

One thing you could do is turn the inside to approximate size and shape, and then take it off the lathe and pour in a cup of sugar as a test. That way you can return it to the lathe, and make minor adjustments. Of course, we probably aren't sure that the actual volume going into this cup will be exactly a cupful. Has someone actually divided this out like a receipe?

This is easier than you think. Notice that d squared x h is also the rectangular solid that the parabola is contained within. A 4" diameter x 4" high parabola fits perfectly inside a 4" cube. The difference in volume is called by boat makers the prismatice coeficient. That difference is expressed here: (pi/8) = .39 or simply 40%. So, for a parabola that is contained within, say, a 3" x 3" x 4" space, just take 40% of that volume, or 36 x .40 = 14.4, which happens to be the volume you are looking for!

This prismatic coeficient for a parabola is a constant .39 It holds true regardless of the shallowness or steepness of the parabola because there is only one parabola. It is just that sometimes you are using the shallow tip and othertimes the steeper portion further up.

Hey, guys, I thought we got out of woodworking and into turning because we weren't good at measuring things!

Bill

Leo wrote: Sorry, I guess this has wandered pretty far OT, but I couldn't resist. ^^^^^^^^^^^^^^^ Bill Rubenstein wrote: Hey, guys, I thought we got out of woodworking and into turning because we weren't good at measuring things! ^^^^^^^^^^^^ Leo writes: Bill, is that a "subtle" way of saying it is, indeed, too far OT?

Leo:

No, not really. I just thought it was turning into a funny thread, considering...

Bill

_____ American Association of Woodturners Cascade Woodturners Assoc., Portland, Oregon Northwest Woodturners, Tigard, Oregon _____

Thanks Leo, Alan, Bill and mat for the comments, info and hints. I think we're all pretty much coming up with the "right" answer as to the height, but the difference appears to be the conversion from gallons to cubic inches. I used the "dry" conversion such that 1 gal.= 268.8 cu in. and yields 4.6". (Anyone know why there is a difference?)

I had planned on hollowing first and pulling it off the lathe to test with some sugar or rice. My real need for the appx. height was due to having to glue up some ebony and cherry to form the blank and wanted to make sure I had enough. I plan on ammonia fuming the box prior to applying finish and will post pics of the finished piece.

_____ American Association of Woodturners Cascade Woodturners Assoc., Portland, Oregon Northwest Woodturners, Tigard, Oregon _____