December 8, 2003
A: FROM MENTOR UTPALA DUBEY IN IL
The center of gravity is a geometric property of any object. The center of
gravity is the average location of the weight of an object. We can
completely describe the motion of any object through space in terms of the
translation of the center of gravity of the object from one place to
another, and the rotation of the object about its center of gravity if it is
free to rotate. If the object is confined to rotate about some other point,
like a hinge, we can still describe its motion. In flight, both airplanes
and rockets rotate about their centers of gravity. A kite, on the other
hand, rotates about the bridle point . But the trim of a kite still depends
on the location of the center of gravity relative to the bridle point,
because for every object the weight always acts through the center of
gravity.
Determining the center of gravity is very important for any flying object.
How do engineers determine the location of the center of gravity for an
aircraft which they are designing?
In general, determining the center of gravity (cg) is a complicated
procedure because the mass (and weight) may not be uniformly distributed
throughout the object. The general case requires the use of calculus which
we will discuss at the bottom of this page. If the mass is uniformly
distributed, the problem is greatly simplified. If the object has a line (or
plane) of symmetry, the cg lies on the line of symmetry. For a solid block
of uniform material, the center of gravity is simply at the average location
of the physical dimensions. (For a rectangular block, 50 X 20 X 10, the
center of gravity is at the point (25,10, 5) ). For a triangle of height h,
the cg is at h/3, and for a semi-circle of radius r, the cg is at
(4*r/(3*pi)) where pi is ratio of the circumference of the circle to the
diameter. There are tables of the location of the center of gravity for many
simple shapes in math and science books. The tables were generated by using
the equation from calculus shown on the slide.
For a general shaped object, there is a simple mechanical way to determine
the center of gravity:
If we just balance the object using a string or an edge, the point at which
the object is balanced is the center of gravity. (Just like balancing a
pencil on your finger!)
Another, more complicated way, is a two step method shown on the slide. In
Step 1, you hang the object from any point and you drop a weighted string
from the same point. Draw a line on the object along the string. For Step 2,
repeat the procedure from another point on the object You now have two lines
drawn on the object which intersect. The center of gravity is the point
where the lines intersect. This procedure works well for irregularly shaped
objects that are hard to balance.

Source: NASA
see also http://www.grc.nasa.gov/WWW/K-12/airplane/cg.slide.html
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A: FROM MENTOR MOLLY WILLIAMS IN MI
The center of gravity of any object is a point at which you could assume the entire weight of the object is concentrated. If you suspend an object, its center of gravity will always hang directly underneath the support (once the object stops swinging and comes to rest). So, here's a practical way to find the center of gravity for a 2-D object. You hang the object by one of its corners (or any point near one edge) so that it can swing freely, and at the same time hang a plumb line (a string with a weight on its bottom end) so that both are suspended from the same point. Trace onto the object the line where the plumb line touches it. The center of gravity has to be somewhere on that line. Now, suspend the object from a different corner and trace the new line from the plumb line. If you do this several times, all of the plumb line tracings should intersect at a single point * the center of gravity. You can double check this result with a final test * you should be able to lay the 2-D object horizontally and balance it on your fingertip by supporting it exactly at the center of gravity. Note that sometimes the center of gravity isn't even on the object, so you couldn't do the fingertip test. For example if the object is a ring, its center of gravity will be at the center of the circle, even though that's empty space.