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Center of Gravity Math Experiment

The term "center of gravity" is often used interchangeably with the term "center of mass" when the objects do not have other forces attracting them to one another, and the gravitation field remains constant. When the center of mass is located, almost any object, regardless of its shape, can be balanced. If an object is uniform, like a ruler, the center of mass will be located in the center of it. If the object is irregular, the center of mass will be located closer to the heavier end; for example, the center of mass of a broom will be closer to the end with the bristles. A common physics experiment to illustrate how to find the center of mass is to find the center of mass of a broom.

Center of Masss Demonstratrion
- Take one finger from each hand and balance the broom so that the broom is laying horizontally on your fingers. At the same time, slowly slide each finger until your fingers touch. The point at which they touch is the center of mass, and the broom should be balanced on your fingers.
Mathematical Expression
- The weight on the right finger (Wr) or left finger (Wl) is equal to the weight of the broom (Wb) multiplied by a fraction containing the distance of the right side (Dr) or left side (D1) over the total distance, which is the sum of the distance on the left and right (D1 + Dr).
  
  To summarize this expression using variables: Wl = Wb * Dr / (Dl + Dr), and Wr = Wb * Dl (Dl + Dr).
Torque
- The broom has a tendency to rotate around each finger if you try to find the center of mass using only one finger. By using two fingers and keeping them level, with the broom held horizontally on top of them, the broom is not able to rotate in either direction, canceling out the any forces related to torque.
Friction
- There is a frictional force between the broom and your fingers. As you begin to move your fingers together, the finger closest to the heavier bristle end will not move at first. This finger is bound by the force of friction. The amount of friction remains constant throughout the surface of the broom, and occurs in opposite directions. When the distance from one finger to the center of mass changes, the weight supported by that finger is transferred to the other finger, allowing the frictional forces to be overcome, and also allowing the finger that is on the heavier end of the broom to slide toward the center of mass.

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