Double Integral Application: Center of Mass (MAT455)

Center of Mass

In this section we are going to find the center of mass or centroid of a thin plate with uniform density ρ.  The center of mass or centroid of a region is the point in which the region will be perfectly balanced horizontally if suspended from that point.

So, let’s suppose that the plate is the region bounded by the two curves  and  on the interval [a,b].  So, we want to find the center of mass of the region below.

We’ll first need the mass of this plate.  The mass is,

Next we’ll need the moments of the region.  There are two moments, denoted by M_x and M_y.  The moments measure the tendency of the region to rotate about the x and y-axis respectively.  The moments are given by,

Equations of Moments

The coordinates of the center of mass, , are then,

Center of Mass Coordinates

where,

Note that the density, ρ, of the plate cancels out and so isn’t really needed.

Let’s work a couple of examples.

Example 1  Determine the center of mass for the region bounded by ,  on the interval . Solution Here is a sketch of the region with the center of mass denoted with a dot. Let’s first get the area of the region.                                                           Now, the moments (without density since it will just drop out) are,    The coordinates of the center of mass are then,                                                                Again, note that we didn’t put in the density since it will cancel out. So, the center of mass for this region is .

Example 2  Determine the center of mass for the region bounded by  and . Solution The two curves intersect at  and  and here is a sketch of the region with the center of mass marked with a box. We’ll first get the area of the region.                                                           Now the moments, again without density, are                       The coordinates of the center of mass is then,                                                               The coordinates of the center of mass are then, . Demonstration of Center of Mass of Lamina Introduction of Center of Mass Example : Center of Mass Example 2: Center of Mass of Lamina