HPP Activity A24.v1
Center of Mass
Exploration
Observe the "Axe Throw" video.
GE 1.
1. Does the axe undergo projectile motion?
Observe the video again. Follow the white spot near the axe head. Record its
position on each frame during the throw.
2. How does the motion of the white spot compare to the simple projectile
motion of a ball tossed in the air?
The total motion of the axe appears complex. If we imagine all the mass concentrated at the
white spot, the motion can be described very simply. All extended objects have such a point
associated with them that move in a relatively simple way. This point is called the center of
mass.
GE 2.
1. What do you think determines the location of the center of mass?
You can use gravity to help determine the center of mass (as long as the
gravitational field is uniform). Just find the balance point.
2. Make a guess for where the center of mass occurs in each of the following
shapes. Mark your guess with a circle.
Light
ball
Double
thickness
Activity Guide
 2010 The Humanized Physics Project
Supported in part by NSF-CCLI Program under grants DUE #00-88712 and DUE #00-88780
HPP Activity A24.v1
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3. Find the center of mass experimentally and mark it in the figure with a
square. How were your predictions?
Invention
Now that you have a intuitive feel for center of mass, let's define it mathematically. Consider a
simple two mass system, as shown below.
x
x
x
xcm
x
m1
x
x
m2
x1
x
x2
For two masses m1 and m2 that are at positions x1 and x2 on the x-axis, respectively, the x
coordinate of the center of mass is given by the equation
xcm 
(m1 x1  m2 x 2 )
M
where M is the total mass of the system.
M  m1  m2
If the masses are not both along the x-axis there would be a similar equation for the y-coordinate
of the center of mass:
(m1 y1  m2 y 2 )
M
Calculations of the center of mass for an extended mass object such as the triangle or circle
require calculus.
y cm 
Application
Activity Guide
 2010 The Humanized Physics Project
HPP Activity A24.v1
Consider the two and three ball objects. Define the x-axis to be along the long rod. You can put
the origin at the center of one of the ball's on the end.
GE 3.
Two Ball System
1. Measure the x-coordinates and masses of the balls.
2. Calculate the position of the center of mass.
3. Compare the theoretical center of mass with the actual center of mass. Are
they the same point?
Three Ball System
4. Measure the x-coordinates and masses of the balls.
5. Calculate the position of the center of mass.
6. Compare the theoretical center of mass with the actual center of mass. Are
they the same point?
GE 4.
1. Does the center of mass need to be within the object?
Consider the following object:
2. Where do you think the center of mass should be? Mark it with a circle.
Activity Guide
 2010 The Humanized Physics Project
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HPP Activity A24.v1
3. Locate the center of mass experimentally. How was your prediction?
GE 5.
Pick a human test subject (the experiment will not be painful). The subject
should stand up straight.
1. Where do you think the subject's center of mass is?
Use a board on a pivot to measure the actual center of mass location.
2. Record the position as a distance measured from the feet.
3. What happens to the center of mass position if the subject bends forwards?
Activity Guide
 2010 The Humanized Physics Project
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