"pipes" is a list of pairs of 3-dimensional coordinates (x,y,z), each pair representing the ends of the 6, 9 or 24 inch pieces of PVC pipe; 
include the welding rod we use as an axle for the throwing arm in the list of pipes

"pointmasses" is a list of 3-dimensional coordinates for each part treated as a point mass (which is everything else)

Use the first plot to make sure that you haven't forgotten anything.

You can ignore the shoe strings and include the cloth sling with the mass of the projectile.

I chose my coordinate origin to be directly under the center of the axle.

pipes={{{-3,-3,0},{3,-3,0}}, (* six inch pipe *)
{{-3,3,0},{3,3,0}}, (* six inch pipe *)
{{-3,-3,0},{-3,-3,9}}, (* nine inch pipe *)
{{3,-3,0},{3,-3,9}}, (* nine inch pipe *)
{{-3,3,0},{-3,3,9}}, (* nine inch pipe *)
{{3,3,0},{3,3,9}}, (* nine inch pipe *)
{{-3,-3,9},{-3,3,9}}, (* six inch pipe *)
{{3,-3,9},{3,3,9}}, (* six inch pipe *)
{{-3,0,9},{3,0,9}}, (* axle *)
{{0,0,1.5},{0,0,25.5}}}; (* throwing arm *)
pointmasses={{-3,-3,0}, (* 1/2 inch tee *)
{3,-3,0}, (* 1/2 inch tee *)
{-3,3,0}, (* 1/2 inch tee *)
{3,3,0}, (* 1/2 inch tee *)
{-3,-3,9}, (* 90 degree elbow *)
{3,-3,9}, (* 90 degree elbow *)
{-3,3,9}, (* 90 degree elbow *)
{3,3,9}, (* 90 degree elbow *)
{0,0,1.5}, (* 1 1/4 by 1/2 inch tee + counterweight *)
{0,0,5.5}}; (* projectile and sling *)
Show[Graphics3D[Flatten[{
Table[Line[pipes[[i]]],{i,Length[pipes]}],RGBColor[1,0,0],PointSize[.04],Table[Point[pointmasses[[i]]],{i,Length[pointmasses]}]}]],Boxed->False]

"masses" is the list of masses of each part, in the order in which they appear in "pipes" and then in "pointmasses"; mtot is the total mass

masses={36,36,54,54,54,54,36,36,28,144, (* pipes *)
26,26,26,26,20,20,20,20,1082,80+6}; (* pointmasses *)
mtot=Sum[masses[[i]],{i,Length[masses]}]

centers=Table[Mean[pipes[[i]]],{i,Length[pipes]}];
bits=Join[centers,pointmasses];
cmass=N[Sum[masses[[i]]*bits[[i]],{i,Length[masses]}]/mtot]

Length[bits]
Length[masses]

This shows the plot again, including the center of mass in blue:

Show[Graphics3D[Flatten[{Table[Line[pipes[[i]]],{i,Length[pipes]}],RGBColor[1,0,0],PointSize[.04],Table[Point[pointmasses[[i]]],{i,Length[pointmasses]}],RGBColor[0,0,1],Point[cmass]}]],Boxed->False]