Centroids and Centers of Mass

Purpose

The purpose of this lab is to acquaint you with the use of double and triple integrals to determine centers of mass and centroids of regions in three dimensions.

Background

In designing mechanisms or structures, one often has to deal with distributed forces, that is, forces that do not act at a discrete, finite set of points. The most common example of a distributed force is the force of gravity, which acts on all parts of any body of matter. Other examples are pressure in fluids and electrostatic forces, though there are many others.

One of the basic useful principles of analyzing distributed forces is the idea of replacing them with a single, aggregate force that acts at a single point and is somehow equivalent to the original distributed force. This may not always be possible, but this technique has found great use in engineering and science. As a simple example, suppose we have a solid plate of uniform thickness and density, but irregular shape. Finding the equivalent force is really the problem of finding the point where we could exactly balance the plate. This balance point is often called the center of mass of the body.

For symmetric objects, the balance point is usually easy to find. For example, the balance point of a see-saw is the exact center. Similarly, the balance points for rectangles or circles are the just the geometrical centers. For non-symmetric objects, the answer is not so clear, but it turns out that there is a fairly simple algorithm involving integrals for determining balance points.

To describe the algorithm abstractly, suppose that we have a three-dimensional body occupying a region D in 3. Suppose further, that the density of the body is described by a function defined on D. Then the mass m of our body can be computed by the triple integral

and the coordinates, written , of the center of mass can be shown to be given by the three equations

As an example, consider the unit cube, and suppose, for the sake of simplicity, that is constant and equal to 1. Then the mass is also unity and are given by

Performing the three integrals gives , as expected from symmetry.

Exercises

1. In the case that the density is uniform, show that are independent of and are given by

   

   where V is the volume of the domain D. In this case, the point is called the centroid of the domain D.

2. In class, we treated the case of a plate of uniform thickness whose density could only depend on x and y. Show how the general equations in this lab reduce to the ones we found in class for this special case.
3. Find the centroids of the following geometrical shapes in two dimensions.
   1. A half-disk of radius R.
   2. An equilateral triangle of side L.
   3. A parabolic spandrel. That is, the region D is bounded below by the x axis, above by the curve , and on the right by the vertical line x=a, where k and a are positive constants. Note that your answer will depend on the parameters k and b.
4. Find the centroids of the following geometrical solids in three dimensions.
   1. The portion of the unit sphere in the first octant.
   2. The region bounded by the paraboloid and the x-y plane.
   3. The tetrahedron bounded by the coordinate planes and the plane x+y+z = 1.
5. Find the center of mass of D, if D is the region bounded by , , the x-y plane, and if is given by
   1. 
   2. 
   3.