# Question Video: Finding the Centre of Mass of a Rectangular Shape with a Rectangular Hole Mathematics

Find the coordinates of the center of mass of the figure, which is drawn on a grid of unit squares.

02:40

### Video Transcript

Find the coordinates of the center of mass of the following figure, which is drawn on a grid of unit squares.

The shape that we’re interested in is the green shaded area on this graph here. Since the shape is nonuniform because it has a big hole in the middle, we’ll need to break it up into several uniform shapes and then use the center of mass formula. The center of mass formula allows us to calculate the coordinates of the center of mass for a collection of objects using the mass and location of each individual object. For extended objects like two-dimensional laminae, we define their location as their center of mass. Okay, now we just need to find the shapes that will make our calculation easiest. We can actually do this calculation using only two laminae if we use the negative mass method.

In the negative mass method, we treat laminae with holes as solid laminae with positive mass combined with a lamina the shape of the hole, but with negative mass. When we fix the lamina with negative mass to the lamina with positive mass, the positive and negative masses cancel, giving us the effective result of zero mass or a hole. We then find the center of mass of these two laminae, the positive mass and the negative mass, using the exact same center of mass formula. To apply the negative mass method to our particular shape, we’ll start with a positive mass lamina with the same perimeter as the outer perimeter of the shape that we’re interested in. To account for the hole, we’ll then include a lamina with negative mass with the same perimeter as the inner perimeter of our shape.

As we can see, the shape that we’re interested in is exactly the portion of the positive mass lamina that does not coincide with the negative mass lamina. Before we worry about particular values for these two masses, let’s find the centers of mass of these two laminae. Recall that for any uniform rectangular lamina, the center of mass is exactly at the point where the two diagonals meet. So to find the centers of mass of these two laminae, we’ll just draw the diagonals. Here, we’ve drawn the two diagonals of the positive mass lamina, and they meet right here. When we include the diagonals of the negative mass lamina, we see that they meet at exactly the same point. So both of these laminae have the same center of mass.

Whenever we combine two objects and those two objects have the same center of mass, the resulting combination also has that center of mass. We can see this mathematically by noting that if all of the 𝑥 sub 𝑖’s have the same value, call it 𝑥, then the numerator is just 𝑥 times the denominator. And 𝑥 times something divided by that something is just 𝑥. Okay, so the center of mass is this center point here. This is four and a half units along the 𝑥-axis and exactly four units up on the 𝑦-axis. Four and one-half is nine-halves. So the center of mass of this figure is located at nine-halves comma four. The ease with which we found this answer shows the power of the negative mass method. Not only were we able to find the centers of mass entirely graphically, we didn’t even need to plug in values into the center of mass formula to get the answer.