College Physics Lab I
PH 141
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Center of Mass

Purpose: To demonstrate the concept of center of mass and its relationship to balancing.

Discussion:
The center of mass is the point at which all of the mass of the system seems to be concentrated for purposes of linear or translational motion.
The quantitative relationship defining the center of mass position (RCM) is

MsysRCM = m1r1 + m2r2 + m3r3 + ... where Msys = m1 + m2 + m3 + ...

RCM is a vector. We can express RCM in terms of its components XCM and YCM:

MsysXCM = m1x1 + m2x2 + m3x3 + ...  and MsysYCM = m1y1 + m2y2 + m3y3 + ...

Since the system behaves as if all of its mass is concentrated at one point, the center of mass, then we should be able to support an extended object in equilibrium through the application of a single upward force (F1) applied at the center of mass. The entire weight (Wsys) of the object seems to act at the center of mass pulling it down. A single upward force should be sufficient to balance the object in equilibrium (aCM = 0).

SF = MsysaCM

F1 - Wsys = 0

And if you observe an extended object in equilibrium supported by a single force, that force must be applied at the center of mass.

Activity 1: Determine the center of mass for a one-dimensional system.
Procedure:

1. Take a meter stick and support it at the 50 cm mark.

2. Hang 4 masses at different positions along the meter stick. Vary the masses and positions until you have balanced the meter stick again.

3. Measure the positions from one end of the meter stick. Measure the position of the support point from the same end of the meter stick. Measure the masses.

4. Using the center of mass relationship predict the center of mass position of the 4 masses:

5. Calculate a % difference between the predicted and observed center of mass positions. Is there good agreement between the values?
 

 Activity 2:  Determine the center of mass for a two-dimensional system.

Procedure:

1. Balance a tray on the end of a bar. Using a water soluble pen mark the center of mass of the tray.

2. Take 3 masses. Place them at different positions on the tray. They shouldn't be too close to center and they shouldn't all lie along one line. Vary the masses and/or move them around until the tray is once again in equilibrium (or as close as reasonably possible).

3. With the water soluble pen mark the position of each mass. You want to be able to find the center position for each mass.

4. First make a coordinate system for your tray. And draw in the displacements to your masses.
a. Pick an point near the lower left hand corner of the tray. Draw one axis parallel to the long side of the tray and the other axis parallel to the short side of the tray.  Then from the origin of your coordinate system draw a line to the observed center of mass and to each of the three masses.

OR

b. Pick the observed center of mass as the origin of your coordinate system. Draw one axis parallel to the long side of the tray and the other axis parallel to the short side of the tray (and thus perpendicular to the long side of the tray).
Then from the origin of your coordinate system draw a line to each of the three masses.

5. For each mass and the center of mass determine the distance and direction (angle) from the origin of your coordinate system.

6. From this information determine the x and y-components of the position of each mass and of the center of mass. Using the center of mass relationships for the x and y directions separately determine XCM and YCM.

7. Calculate a % difference between the predicted and observed center of mass positions for both x and y directions. Is there good agreement between the values?