```THE CENTER OF MASS

Joan C. Preer                  Beasley Academic Center
5255 South State Street
Chicago, IL 60609
1-312-536-8745

Objectives:
1.  To  determine  the  center of mass of two irregularly shaped objects.
2.  To  determine  when  the center of mass is inside the boundaries  of the
object and when the center of mass is outside the boundaries of the object.
3.  To  determine when an object is stable or balanced or why an object is
unstable or unbalanced.

Materials:   8-multi-speed cars
10-forks
10-spoons
10-plastic cups
10-irregularly shaped object cards
10-strings
10-rubber washers
10-nails
1-Leaning Tower of Pisa
1-double cone, cylinder and incline
1-styrofoam center of mass fish
1-3.81 cm metal sphere
10-boxes of colored chalk
10-Q-tips

Strategy:
Opening:  Toss  a  metal  sphere  across  the  room;  it  travels a  smooth
path  or  parabola.   Then  toss  the styrofoam  fish  across  the  room;  it
does  not  follow a smooth  path.   It  wobbles  all  over the place except   for
a very special place we are going to discuss today.  We are going to talk about
the center of mass.
1.  Where is  most of the mass of the  fish concentrated?
2.  Look at  the two lights as the fish is being tossed through the air.
Does the path of either light form a parabolic curve?

Discussion:  The center of mass is the average position of all the particles
of mass that make up a particular body or object.  The metal sphere is a
symmetrical object and its mass is concentrated at its center. In contrast, more
of the mass of the fish is concentrated toward the head of the fish.  Therefore,
the center of mass of the fish is toward the heavier end or the head of the fish.
When we toss the fish it does wobble all over the place.  However, one of the
lights does follow a parabolic path.  We used this light to mark the center of
mass of the fish.

Activity:  Slide a nail through one of the two holes in the object.  Slip the
string over the opposite end of the nail and let it swing freely using the washer
to weight the string.  Chalk the string letting it mark the straight path, from
the point of suspension to the opposite end of the card.  Then repeat these same
steps with the second hole in the object.  The center of mass will be located
where the two chalked lines intersect.  The center of mass can be checked by
placing the eraser of a pencil at the point of intersection and see if the object
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balances on the pencil.

1.  Why  does  the  object  swing back and forth when placed on the nail?
2.  Where is most of the mass of the object concentrated?
3.  Is  the center  of mass of the object within the boundaries of the
object?

Discussion:  (Sketch two L-shaped figures on the board.)

Figure a               Figure b

If we drop a line straight down from the center of mass of a body of any
shape and it falls inside the base of support, as in Figure a, it is in stable
equilibrium and the object will be balanced.  However, if the center of mass falls
outside of the base of support it is unstable and will not be balanced, as in
Figure b.  In our irregularly shaped object, the center of mass is within the
boundaries of the object.

Activity:  Stand against the wall and try and touch your toes.
1.  Where is your center of mass when you lean forward to touch your toes?
2.  What is the base of support for your body?
3.  Is your center of mass inside or outside your base of support?
4.  Why does your body rotate* forward?
*The rotation of your body as you move forward is called torque.
Then stand away from the wall and try to touch your toes.

1.  After you move away from the wall, where is your center of mass?
2.  Is the center of mass inside or outside your base of support?
3.  This time the body does not rotate forward. Why?

Discussion:  You can lean over and touch your toes without rotating forward
only if your center of mass is above the area of your base  of support.  In this

Activity:  Determine the center of mass of the multi-speed car. Once you have
found the center of mass, select your car's speed.  Then joining forces with
another team, leave one car stationary and run the other car into one side of the
front end of the stationary car.  Then run your car into the rear of the same side
of the stationary car.  Finally, run your car into the center of mass of the
stationary car.  Determine what happens in each case.

1.  Where is the center of mass of the car?
2.  When the stationary car is hit in the front, of one side, in what
direction does the car move?
3.  When the stationary car is hit in the rear, of one side, in what
direction does the car move?

Discussion:  The center of mass of the car is a few millimeters in front of
the rear wheels.  When the stationary car is hit in the front, the car moves
counterclockwise denoting a counterclockwise torque.  When the stationary car is
hit in the rear, the car moves clockwise denoting clockwise torque. When the
stationary car is hit at the center of mass, the counterclockwise and the
clockwise torques are equal.  The net torque is equal to zero and there is no
rotation.  The stationary car is just pushed forward.  In relation to the moving
car, they are perpendicular to one another.

Activity:  Fasten  a  fork, spoon, and one Q-tip together as shown below:

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The combination will balance nicely on the edge of a plastic cup (it may be
necessary to weight the cup).

1.  Where is the center of mass for the whole setup?
2.  Would this work if the Q-tip were shorter?

Discussion: It is possible to balance this combination on the cup because the
center of mass is somewhere below the point of support.  The heavy handles of the
fork and the spoon curve toward the cup.  This shifts the center of mass of the
entire structure to a point directly beneath the spot where the Q-tip rests on the
cup, putting the fork, spoon and Q-tip in a state of stable  equilibrium; it is
balanced.  This will work with a  shorter Q-tip.  However, the Q-tip will have to
be long enough to rest on the rim of the cup and have some of the Q-tip overlap.

Demonstration:  I have two objects I would like to demonstrate.  I want you
to consider everything we have covered up to this point.  After the demonstration
I would like you to explain how you think each object works.

1. Leaning Tower of Pisa: When the top is removed, the center of mass of the
Leaning Tower of Pisa lies above a point of support, and therefore the tower is in
stable equilibrium.  When the top is added, the center of mass is outside the base
of support.  The Tower is unstable so it topples over.

2. The Double Cone and Incline: When the cylinder is placed at the top of the
incline unrestricted, gravity causes it to roll to the bottom, down hill.
However, when the cone is placed on the incline it appears to roll up hill defying
the law of gravity. The center of mass of the cone is concentrated at the center
of the cone and the cone tapers as you reach the ends. When the cone is placed at
the bottom of the incline it is placed there at its center of mass. The center of
mass always seeks the lowest position it can reach. The track of the incline
widens permitting the center of mass of the cone to be lowered.  This allows the
cone to reach a more stable position.  To reach this stable position the cone
rolls up hill.

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