Determine the position of the center of mass of a solid triangular pyramid with vertices at (0; 0; 0), (1; 0; 0), (1; 1; 0),
and (1; 1; 2).
We need to integrate the density of our object (constant, 1) over the volume of our object. If we weight this integral with a position variable (x,y or z) then we get the center of mass coordinate for each coordinate. See attached for details.
We need to solve the following equations to get the center of mass:
First, draw picture.
If we look down at our figure directly from above:
Three of our vertices are located in the plane z = 0. The fourth is indicated by the circle - it is located above the point (1,1).
Now, we know we need to integrate over the volume of our object. When integrating over an object of this type (straight lines define the edges), it is standard practice to using these lines as the limits of integration. To set up the integral in the denominator of our equation, we could use the following:
We start by looking at the innermost ...
This job determines the center of mass coordinate for each coordinate. The solid triangular pyramid with vertices are given.