In the figure shown, find the
position of the center of mass of the uniform triangular lamina 𝐴𝐵𝐶, considering
𝐴 to be the origin point.
In this figure, we see the triangle
with vertices 𝐴, 𝐵, and 𝐶 positioned on this 𝑥𝑦-coordinate plane. We want to find the center of mass
of this triangular lamina, and we knew that that corresponds with the centroid or
geometric center of the shape. Just estimating by eye, we might
put the geometric center of this triangle here. But to know the 𝑥- and
𝑦-coordinates of this point accurately, we’ll call them 𝐶𝑂𝑀 𝑥 and 𝐶𝑂𝑀 𝑦,
we’ll need to recall a more precise approach. Whenever we’re working with
triangles seeking to find their center of mass, the key information for doing this
is the coordinates of the triangle’s three vertices.
If we know these, then regardless
of the shape of the triangle, we can calculate the 𝑥- and 𝑦-coordinates of its
center of mass using these relationships. Basically, they involve solving for
the average 𝑥-coordinate and the average 𝑦-coordinate among the vertices. If we apply these relationships to
our scenario with triangle 𝐴𝐵𝐶, then we can note that the coordinates at the
vertex 𝐵 are zero, five 𝑎; those at vertex 𝐶 are four 𝑎, zero. And because vertex 𝐴 is positioned
at the origin, those coordinates are zero, zero.
To solve then first for the center
of mass 𝑥-coordinate, we’ll add together zero, zero, and four 𝑎 and divide all
that by three, which gives us four 𝑎 over three. And then to solve for the center of
mass 𝑦-coordinate, we’ll add together five 𝑎, zero, and zero, the 𝑦-coordinates
of the three vertices of our triangle, and divide all that by three, giving us
five-thirds 𝑎. These, then, are the coordinates of
the center of mass of this uniform triangular lamina.