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Huge letter billboard


The application of mathematics in life is not only to calculate the addition, subtraction, multiplication, and division of various numbers. It also includes applications to many things in life. For example, when we are building a new building, we have to calculate many things, such as Weighing of the building, calculating the position of the weighing column, and the method of delivering materials to the roof. The concept of center of mass is involved here. The center of mass is the center of the mass of an item. If this center can be found, it will be easier to hoist the item, and at the same time, some dangers (decoupling due to uneven weight) will be avoided. It is very simple to simply transport some steel bars, because the center of mass can be found directly. But for example, for some strange shapes, such as the name of a company, those huge letters, such objects need to be calculated to find its center of mass.

Then we take the letter F as an example, because its center of mass cannot be seen directly, and it is not too complicated, which also helps us to explore.

The length of each side of it is shown in the figure
We also divide him into three, three rectangles, we find their centroids respectively, if the lower left corner is regarded as the origin (0.0), then the three rectangles are called a, b and c respectively.
a is composed of points A, B, J, K. b is composed of points C, D, E, F. c is composed of points G, H, I, J.

Then, I can find the centroids of the three rectangles at a(0.2,1.25),b(0.8,1.3),c(0.9,2.3). So here we need to start some calculations, here is the equation for calculating the centroid:
Because it is a uniform quality, then use the area to express the quality.

The coordinates of the mass point x: (0.4*2.5)*0.2+(0.4*0.8)*0.8+(0.4*1)*0.9=((0.4*2.5)+(0.4*0.8)+(0.4*1))*x.

The coordinates of the mass point y: (0.4*2.5)*1.25+(0.4*0.8)*1.3+(0.4*1)*2.3=((0.4*2.5)+(0.4*0.8)+(0.4*1))*y

Finally, we can find that the coordinates of the center of mass is (0.474,1.503)
It looks like it is on the image, but in fact its center of mass is not on the "F", it is on the outside.
By the way, I gave it a thickness of 0.2, so its true center of mass is (0.474, 1.503, 0.1)
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