X Marks the Spot: Finding the Center of Mass

Summary

Key Concepts

Geometry, gravity, center of mass

Credits

Ben Finio, PhD, Science Buddies

Introduction

You can probably find the center of simple shapes, like circles and squares, pretty easily. But, how do you find the "middle" of an irregular shape, like a drawing of a dog or a cat? This project will show you how to do it using nothing but string and paper clips!

This activity is not recommended for use as a science fair project. Good science fair projects have a stronger focus on controlling variables, taking accurate measurements, and analyzing data. To find a science fair project that is just right for you, browse our library of over 1,200 Science Fair Project Ideas or use the Topic Selection Wizard to get a personalized project recommendation.

Background

How do you define the exact center of an object? One way to do this is to find the object's center of mass. The center of mass is the point about which an object will balance if you try to rest it on your fingertip. Or, if you hang an object, for example a picture frame from a nail, the center of mass will hang directly below the nail.

For symmetrical objects, finding the center of mass is easy. For example, for a rectangular picture frame, you know the center of mass is in the middle of the rectangle, and you can find it with a ruler. When you hang the picture frame, you will make sure it is centered on the nail so it will be level – otherwise it will tip to one side. The same applies to other symmetric objects, like a spherical basketball – you know the center of mass is in the middle of the sphere.

What about irregularly shaped objects, like a dog or a cat or a person? Now finding the center of mass is not so easy! This activity will show you how to find the center of mass for any two-dimensional shape you cut out of paper, using a trick that has to do with the hanging picture frame mentioned above. If you hang a shape from a single point, you know the center of mass will always rest directly below that point. So, if you hang a shape from two different points (one at a time), and draw a line straight down from each point, the center of mass is where those lines intersect. This technique can be used for any irregular 2D shape. Don't believe it? Try this activity to find out!

Materials

Paper (heavier paper, like construction paper, cardstock, or thin cardboard from the side of a cereal box will work best)
Scissors
String
Pencil
Ruler
2 paper clips, or a pushpin and another small, relatively heavy object you can tie to the string (like a metal washer)

Instructions

Cut a piece of string about one foot long, and tie a paper clip to each end. Alternatively, you can use any other small object like a metal washer on one end (this will serve as a weight), and any other small, pointy object like a needle or pushpin on the other end (this will be used to puncture the paper).
Start out with an easy shape: a rectangular piece of paper or cardboard. Can you guess where the center of mass of the rectangle is? If so, use a ruler to measure where you think it will be, and mark this spot with your pencil.
Punch several small holes around the edge of the paper. Make them as close to the edge as possible without ripping the paper (this is important for the accuracy of this technique). The exact location of the holes does not matter, but this technique will work best if you space them all the way around the edge (do not just put two holes right next to each other).
Now, poke one end of one paperclip (or pushpin) through one of the holes to act like a hanging hook. Make sure the paper can swing easily from the hook and does not get stuck (rotate it back and forth a few times to loosen the hole if necessary).
Hold on to your "hook" and hold the paper up against the wall. Let the paper swing freely and make sure the string can hang straight down and does not get stuck.
Use a pencil and ruler to draw a straight line on the paper along the string. Does this line go through the center of mass you predicted earlier?
Now, hang your paper from a different hole and repeat the process. Where does this line intersect the first line?
Repeat the process several more times with different holes. Do all the lines intersect at the same point?
Now, cut out an irregular shape. You can cut out a "blob," or draw something like a dog or cat and then cut out the outline. Make sure the shape you cut out remains stiff and flat (i.e. do not cut very thin sections that might be floppy). Can you use a ruler to predict where the center of mass of your irregular shape will be? This is much harder!
Punch holes around the edge of your irregular shape and repeat the experiment. One at a time, hang the shape and the string from one of the holes, and draw a line along the string. Where do the lines intersect? Does this match up with what you predicted?

Extra: if you use a stiff enough material to cut out your shape (like cardboard), can you try balancing it on your fingertip at the center of mass? What happens if you try to balance it about another point?

Observations and Results

You should have found that the center of mass of the rectangle is right in the middle of the page – halfway along the width, and halfway along the height. You can easily locate this spot with a ruler. Then, when you hang the rectangle from a hole on its edge, the string should always pass through this point, regardless of which hole you use. While it is much harder to predict the center of mass for an irregular shape, the same principle holds true. Regardless of what point you hang the irregular shape from, the string will always pass through the center of mass. So, if you hang it from two or more points (one at a time), you can find the intersection of these lines and that is the center of mass.

Note that, due to small errors in the experiment (like friction on the hook that prevents the paper from rotating perfectly, or the holes not being close enough to the edge of the paper), if you draw multiple lines, they might not all intersect in exactly the same place, but they should still be fairly close to each other.