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Balance point of a solid

• To: mathgroup at smc.vnet.net
• Subject: [mg113523] Balance point of a solid
• From: Andreas <aagas at ix.netcom.com>
• Date: Mon, 1 Nov 2010 05:00:11 -0500 (EST)

Hi everyone,

This isn't strictly a Mathematica question, but once I formulate a way
to think about the problem, I'll use Mathematica to generate solutions
to specific situations.  Besides, this seems like the smartest forum
going.  So...

Background

Start with a equilateral triangle as the base of a solid. The triangle
has sides:

a1 = a2 = a3 = 1

At right angles to each side of this base triangle stands a trapezoid
with heights:

h1 , h2, h3

These heights may have different lengths.   Typically h1 > h2 > h3,
but in some cases any or all of them could have equal values.

Of course, all the h's stand at right angles to the base triangle.

Now connect the points of h1,  h2, and h3 and we have a solid composed
of:

1 equilateral triangle (the base)
3 different trapezoids all with their base lengths =1
1 triangle on top.

I can find the area of each trapezoid like this:

A  =  area
A1 = 1/2 x a1 x (h1 + h3)
A2 = 1/2 x a2 x (h1 + h2)
A3 = 1/2 x a3 x (h2 + h3)

I can find the volume like this:

Volume = V

V = A1 x A2 x A3
or
V = (a1 x a2 x a3 x (h1 + h3) x  (h1 + h2) x (h2 + h3)) / 2^3

Problem:

I need to find the point on the base triangle upon which I can balance
the solid.

This is not the centroid of the base triangle, because the solid would
have more weight or volume towards it's highest side, typically h1.

Simply described, I want to find the balance point on the base
triangle such that if the solid rested on a pin at that point, the
base triangle would remain parallel to the floor.

My guess is that I need to calculate the center of gravity of the
volume, then project a line to the base triangle where it would
intersect the base triangle at a right angle.  Basically just dropping
the center of gravity to the bottom plane.

Maybe a bit tedious, but I should have all the information to
calculate this.  Given the right angles and known lengths I think I
have enough information to calculate all angles and lengths of the
solid.

Will this work?

How would I calculate the center gravity for the solid described?

Follow up question...

If this solves the problem for a 3 dimensional solid of this nature,
how can I extend the solution to a 4th, 5th, or nth dimensional
space?

Thanks to all.

A

• Follow-Ups:
  • Output x^-1
    • From: "Dave Snead" <dsnead6@charter.net>