SpudFiles

My Calc 1 class is going to be making models of the 3-dimensional shapes produced when curves are rotated about the x-axis as part of a unit on finding the volumes of such shapes.

I plan to model the shape produced by y=x^(1/2) starting from x=0 to whatever upper boundary is the square of the radius of the material I find. I will need to make a template scaled such that one graphical unit equals some measurement, careful to make the scale small enough that the curve is easily recognizable. Afterwards, I'll make a mold of the shape that can be filled with water to validate our calculation of its volume.

I have a lathe but it is not quite operational yet. If I can get it working I'll make the shape out of wood. If not, I'll have to mount some softer material in my drill press and turn it down like Technician did with his QDV pistons.

My first question is: what material should I use and how should I mount it to a 1/2" chuck? Styrofoam was the first thing to come to mind and I considered drilling a hole partially through a piece and gluing a dowel into it. I have a feeling that one slip with sandpaper will cause the dowel to shear right out of the hole or at least damage the foam. A heavier material wouldn't be as fragile, but it would be harder to shape into a curve using a drill press.

My second question is: What curves do French curves actually contain? It would be convenient to use a French curve to gauge the shape of my workpiece as I already have several, but I don't know the equations (if any) of the curves they contain. Other than sitting down with my calipers to make an x&y table to figure out the equations, I have no way of knowing. The internet hasn't been very helpful either. (And the fact that there are many types...)

Lastly, would I be wasting my time trying to make a shape like this? Are there any ready-made cones produced by specific equations that I can just buy? I know rocket nose cones come in all different varieties, especially parabolic.

All input is appreciated as always.

Laminated wood.

It's cheap. It's easy to work with.

And it's easy to chuck into a lathe with nothing more than a flange and some plumbing pipe.

Also.... Rather than make a "negative" image that you can fill with water, how about just a "positive" image immersed in water (measure displacement)?

As for French curves, I've no idea... Splines would be my guess.

As for nosecones... Few are parabolic. Most are either elliptical (which admittedly is quite similar to parabolic since a parabola is just an ellipse of infinite major axis) or ogives (of which there are many different flavors... tangent ogives probably being the most common in the hobby world with another (Von Karmen??) being more common in the professional world).

I've never worked personally with a lathe but I hope I can help.
Print out a picture of the graph on regular paper, and then transfer it to some slightly thick sheet metal ( 1/4" ) and cut out the shape. Mount that onto the tool rest of your lathe and then (making sure it's parallel to the material you are using), use it as a physical guide for the tool. once you've cut it once, shift the metal piece a little ways forward, and repeat the process until the thinnest part (x=0) will guide your tool to cut all the way through the material, and then stop cutting & remove the piece.

This'll make a 'lathed' version of the graph. Basically it might look like a nose cone to a rocket or a blunt-nose bullet, again I'm not sure if this is what you were looking for but I hope it helped.