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.
