Centroids
A centroid is the center of gravity or the center of mass of an object. An object is in a state of equilibrium if balanced along it’s centroid. So if you took a very thin and strong needle, and placed it under an object’s centoid, the object will be balanced. So how could you find the centroid of an object?
These equations appear to be complex, but in fact, they are very simple. What the letters with the overline mean is the center of the y-direction, the center of the x-direction, and the center of the z-direction. This equals the sum of the area(S) multiplied by the center(s) of the y, x, or z-directions and that’s divided by the sum of the area(s). So what does that mean exactly?
It’s simple, let’s start with a cube. First we need to choose a corner (everything we measure will be measured from that corner), and then we need to solve for the equations. Let’s say the cube is 2 inches by 2 inches by 2 inches, and we chose a corner on the lower left hand side. From the corner, the center of the cube in the x-direction is 1 inch. If we had a cube with a semicircle on the right side, we would measure the center of the semicircle from the corner, in the x-direction (but we don’t have one in this example). With this value (1 inch) we would multiply it by the area of a face of the cube. So it would be 2 inch X 2 inch=4 inches. Four inches multiplied by the 1 inch=4inch. After that we would divide it by the sum of the area (in this case we only have one shape). So 4 inches divided by 4 inches=1 inch, so 1 inch would be our x-direction centroid. After that we would do the same thing for the other two centroids.
If centroids still don’t make sense to you, I still have another example. In this next example, the shape I chose was a “T” shape. The horizontal or top part of the “T” is 10cm long and 2cm tall. The vertical part is 1cm wide and 10cm tall. Since the shape is only 2D, we’ll use the two equations:
It’s simple, let’s start with a cube. First we need to choose a corner (everything we measure will be measured from that corner), and then we need to solve for the equations. Let’s say the cube is 2 inches by 2 inches by 2 inches, and we chose a corner on the lower left hand side. From the corner, the center of the cube in the x-direction is 1 inch. If we had a cube with a semicircle on the right side, we would measure the center of the semicircle from the corner, in the x-direction (but we don’t have one in this example). With this value (1 inch) we would multiply it by the area of a face of the cube. So it would be 2 inch X 2 inch=4 inches. Four inches multiplied by the 1 inch=4inch. After that we would divide it by the sum of the area (in this case we only have one shape). So 4 inches divided by 4 inches=1 inch, so 1 inch would be our x-direction centroid. After that we would do the same thing for the other two centroids.
If centroids still don’t make sense to you, I still have another example. In this next example, the shape I chose was a “T” shape. The horizontal or top part of the “T” is 10cm long and 2cm tall. The vertical part is 1cm wide and 10cm tall. Since the shape is only 2D, we’ll use the two equations:
It’s easier to solve for a shape’s centroid if you draw it into smaller geometric shapes and solve for their centroids (like I mentioned earlier). So we’ll make the “T” into the top shape and the bottom shape (both are rectangles).
Below are the calculations for the centroids, it would help making a table to make it more organized and easier to read. What the columns are is the x/y-direction centroid for each shape, the area for each shape, and the area multiplied by the centoid. Then area for each shape is added together and the area multiplied by the x/y-direction centroid is added together. |
What I did above is listed the centroids for the two rectangles that make up the "T" shape. (There are different calculations for different shapes. For example, if there was a right traingles cut out of the shape, then the x/y-directions for the shape would be the base of the triangle divided by 3 and the height of the triangled divided by 3, remembering that the right triangle doesn't exist and would be negative in your calculations. If you had a semicircle, the radius would be the centroid to one of the directions because it's half of the circle, and then the other centroid would be four multiplied by the radius and then divided by the quantity three multiplied by pi.)
Using the calculations above, we then place them into their formulas:
Using the calculations above, we then place them into their formulas:
So the centroid for the “T” shape is 5cm in the x-direction and 9cm in the y-direction from the lower left. So what I
did is cut the shape out of cardboard and placed a needle under where it’s centroid should be.
did is cut the shape out of cardboard and placed a needle under where it’s centroid should be.
As you can see, the cardboard is perfectly balanced when the tip of my dull pencil is beneath the centroid.