Before discussing center of gravity, let's assume a few parameters. One, that you're dealing with an object that's on the Earth's surface, not out in space somewhere. And two, that the object is reasonably small – say, not a spaceship that's parked on Earth, waiting to take off. Once all those extraterrestrial influences are eliminated, you're in a fine position to calculate center of gravity for geometric objects using a relatively simple formula – and in fact, because of those conditions just set, you'll use the same formula to find the center of gravity as to find the center of mass.
How to Write About Center of Gravity
Center of gravity in a two-dimensional plane is usually denoted by the coordinates (xcg,ycg) or sometimes by the variables x and y with a bar over them. Also, the term "center of gravity" is sometimes abbreviated to cg.
How to Calculate CG of a Triangle
Your math or physics textbook will often have charts in it for determining the center of balance of certain figures. But for some common geometric shapes, you can use the appropriate center of gravity formula to find that shape's center of gravity.
For triangles, the center of gravity sits at the point where all three medians intersect. If you start at one vertex of the triangle and then draw a straight line to the midpoint of the other side, that's one median. Do the same for the other two vertices, and the point where all three medians intersect is the triangle's center of gravity.
And of course, there's a formula for that. If the coordinates of the triangle's center of gravity are (xcg,ycg), you find its coordinates thusly:
Where (x1,y1), (x2,y2) and (x3,y3) are the coordinates of the triangle's three vertices. You get to choose which vertex is assigned which number.
Center of Gravity Formula for a Rectangle
Did you notice that to find the center of gravity for a triangle, you just average the value of the x-coordinates, then average the value of the y-coordinates, and use the two results as the coordinates for your center of gravity?
To find the center of gravity for a rectangle, you do exactly the same thing. But to make your calculations even easier, assume that the rectangle is oriented squarely to a Cartesian coordinate plane (so it's not set at an angle), and that its lower left vertex is at the origin of the graph. In that case, to find (xcg,ycg) for a rectangle, all you have to calculate is:
If you don't want to relocate your rectangle to the origin of the coordinate plane or if for whatever reason it's not exactly square to the coordinate axes, you can face this slightly scarier-looking, but still effective, formula to average all its x-coordinates to find the value of xcg, and average all the y-coordinates to find the value of ycg:
The Center of Gravity Equation
What if you need to calculate center of gravity for a shape that fits all the assumptions first mentioned (basically, you're not trying to do literal rocket science by finding the center of gravity for objects out in space), but it doesn't fall into any of the categories just mentioned or into the charts in the back of your textbook? Then you can subdivide your shape into more familiar shapes, and use the following equations to find their collective center of gravity:
Or to put it another way, xcg equals the area of section 1 times its location on the x-axis, added to the area of section 2 times its location, and so on until you've added up the area times location of all sections; then divide that entire amount by the total area of all sections. Then do the same for y.
Q: How do I find the area of each section? Dividing your complex or irregular shape into more familiar polygons lets you use standardized formulas to find area. For example, if you've divided that shape into rectangular pieces, you can use the formula length × width to find the area of each piece.
Q: What's the "location" of each section? The location of each section is the appropriate coordinate from that section's center of gravity. So if you want y2 (the location for segment 2), you actually need to provide the y-coordinate for that segment's center of gravity. Again, this is why you subdivide a weirdly shaped object into more familiar shapes, because you can use the formulas already discussed to find each shape's center of gravity, and then extract the appropriate coordinate(s).
Q: Where does my shape go on the coordinate plane? You get to choose where your shape sits on the coordinate plane – just keep in mind that your answer's center of gravity will be in relation to the same point of reference. It's easiest to place your object in the first quadrant of your graph, with its bottom edge against the x-axis and the left edge against the y-axis so that all x- and y-values are positive, but also small enough to be manageable.
Tricks for Finding the Center of Gravity
If you're dealing with a single object, intuition and a little logic are sometimes all you need to find its center of gravity. For example, if you're considering a flat disk, the center of gravity will be the center of the disk. In an cylinder, it's the midpoint on the cylinder's axis. For a rectangle (or square), it's the point where the diagonal lines converge.
You might have noticed a pattern here: If the object in question has a line of symmetry, the center of gravity will be on that line. And if it has multiple axes of symmetry, the center of gravity will be where those axes intersect.
Finally, if you're trying to find the center of gravity for a truly complex object, you have two options: Either whip out your best calculus integrals (see Resources for a triple integral that represents the center of gravity for a non-uniform mass) or input your data into a purpose-built center-of-gravity calculator. (See Resources for an example of a center-of-gravity calculator for radio-controlled planes.)