PhysicsLAB Resource Lesson
Center of Mass

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The center of mass is the point where all of the mass of the object is concentrated. When an object is supported at its center of mass there is no net torque acting on the body and it will remain in static equilibrium. An easy way to determine the location of the center of mass of a rigid pole is to support the pole horizontally on one finger from each hand. Gently slide your fingers together. When your fingers meet, you will be at the center of mass at which time you can easily hold up the pole with only one finger as long as it can withstand the entire weight of the pole. Try it with a bat or a broom. If the object is uniform, for example a meter stick, the center of mass will be at the exact geometric center; if the object is irregular in shape the center of mass will be closer to the heavier end.
 
 
Another method of finding the center of mass of an planar object is through the use of a plumb line. Suspend the mass from each vertex and trace the plumb line's location. Since the center of mass will fall below the suspension point (in order to reduce any torques from the object's weight) the center of mass will be at the intersection of all of the plumb lines.
 
 
If the center of mass is a point within the object's actual structure, then the object can be balanced at that point. The object will also be free to rotate about that point. Consider a uniform rectangular plane. Its center of mass would be at its geometric center. When supported at that point, the mass would remain at equilibrium or would spin uniformly. As the rectangle spins, you can mentally note the location of its center of mass - the only fixed, stationary point in the animation. (press F5 to refresh)
 
 
If a rigid body is projected through the air, its center of mass will follow a natural parabolic arc. For example, consider a baseball bat slung into the air after striking the baseball. Although it appears to have a complicated motion if you watch it rotate, the trajectory will be parabolic if you track its center of mass. The same for a baton tossed up into the air by a majorette. Although its ends are spinning rapidly as it climbs and falls in the air, its path is that of a simple projectile thrown straight upwards. Another example is from one from ballet. The following diagram and commentary of the physics involved when a ballerina completes a grande jete is courtesy of Lenox Software.
 
"Her head stays at nearly a constant height above the floor, giving the illusion of floating across the stage. Inputting the relative masses of her body parts and locating them on each frame shows the path of her center of mass to be a parabola."
 
In most cases the center of mass of an object is a point with physical mass, in other instances it can be located at a position that has no "physical mass"; for example, the center of a ring (a donut) or the center of mass of a boomerang. If the object is irregular in shape, the center of mass is always located closer to the more massive end. However, as long as a "plumb line" dropped from the center of mass falls within the area of an object's base of support, an object will not topple - for example:
 
 
The degree of stability in an object's position depends on how must its center of gravity will be changed if it is moved. Consider the three right circular cones shown below. The left cone is unstable since its center of gravity is lowered if  the cone is tilted in any direction. The center cone is stable since work must be done to raise the center or gravity if the cone is to be tipped. The right cone is neutral since rolling the cone along the surface of the table does not raise or lower its center of gravity.
 

The terms "center of mass" and "center of gravity" are interchangeable as long as their is
no discernible difference in the pull of gravity from one part of the object to another.
 
Many types of art work, magic stunts, and toys use the fact that an object will return to stable equilibrium as long as the system's center of gravity passes within the base of support or below the pivot point. Initially, consider Alexander Calder's kinetic sculptures, or mobiles. At first Calder incorporated motors to keep his sculptures moving, but eventually he figured out ways to allow the careful balancing of the parts to result in their  being moved simply by the wind.
 
When built, Spiral was "the largest mobile in Europe because of its 10 meter height. This piece is made of black steel and weighs two tons. Each arrow is 7m long. The mobile presents a harmonious figure as the five balanced arms are put into motion with the slightest gust of wind."
 
"Spiral" (1958) Alexander Calder
 UNESCO Art Gallery

"Another mobile, 'Steel Fish' (1934), involves an intricate system of weights and balances, and depends on the strength of the wind to arrange or rearrange its composition."
 
But perhaps Calder's most famous mobile is the one on display in the National Gallery of Art in Washington, DC.
 
The stability of balancing toys is built in by constructing them so that their center of gravity always remains below the pivot point. If the toys are tipped in any direction the center of gravity is raised. This results in gravity exerting a restoring force (actually a torque) which pulls it back towards an upright position.
 
 
 
Picture of Balancing Man As long as the center of gravity is below the pivot point, an object will remain in stable equilibrium, even when pushed "off -center." The center of gravity of this toy called "Billy, the Balancing Man" was found by tying a plumb line to one side of the toy, and then allowing it to hang in equilibrium. The center of gravity is marked with an "x," a distance equaling 2.1 centimeters below the pivot point. Note two features:
 
(1) the symmetry of the toy dictates that its center of gravity must lie on a line along the symmetry axis, that is on a line along the body of man; and
 
(2) like the triangular figures shown above, the center of gravity must lie somewhere along the string.
 
 
Watch these videos of the toy as it oscillates about its pivot point either in a mode parallel or perpendicular to the plane of the toy. Note that the toy always remains in stable equilibrium. Videos courtesy of Physics Fun at Clemson University.
 
Picture of Balancing Man
 
A common magic trick using the properties of center of gravity is shown below. The two forks are balanced on the edge of the glass by a toothpick.
 

"When you try to balance an object, if the point of support, the pivot point, is not at the center of gravity then the object will rotate either clockwise or anti-clockwise depending on which side has more torque. However, if the pivot point is on the same vertical line as the center of gravity, then the object, no matter what shape, is going to balance. It will be stable if the center of gravity lies below the pivot point. If the center of gravity is above the pivot point, even a slight disturbance will pull it off balance. In our case if you want to have a stable situation, the center of gravity of this assembly has to be below the pivot point. The pivot point is where the toothpick rests on the rim of the glass. The actual center of gravity must lie in the empty space between the two forks and below the pivot point to achieve stability."

Now try flying a toy bird on the tip of your finger!
 

picture courtesy of
Arbor Scientific
 
 
 
 
Bibliography:

Arbor Scientific - Balancing Bird
Tsing Barden - Physics Toys - Center of Gravity of Two Forks
Cornerstone Networks - Lesson 15 - Calder Mobile
Lenox Software - ballerina image
Chris Reeder Video - www.unicyclist.com
John Regehr - Calder image in the National Gallery of Art
San Francisco Museum of Modern Art - Calder "Steel Fish" (1934)
TradingStories.com - balance toys
Dr. Raymond C. Turner - Clemson University - Billy, the Balancing Man
UNESCO - Calder "Spiral" (1958)



 
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