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.

If the mass of the object is not uniformly distributed, we can characterize the mass distribution by a function w(x) which indicates that the weight is some function of distance x from a reference line. If we can determine the form of the function, there are methods to perform a calculus integration of the equation. We will use the symbols "S[ ]dx" to denote the integration of a continuous function. Then the center of gravity can be determined from:

cg = (S[x * w(x)]dx) / (S[w(x)]dx)

If we don't know the actual functional form,
we can numerically integrate the equation using a spreadsheet
by dividing the distance into a number of small distance segments and
determining the average value of the **mass** (or weight) over
that small segment. Taking the sum of the average value times the distance times the distance segment
divided by the sum of the average value times the distance segment will produce
the center of gravity.

Navigation..

- Beginner's Guide to Aerodynamics
- Beginner's Guide to Propulsion
- Beginner's Guide to Model Rockets
- Beginner's Guide to Kites
- Beginner's Guide to Aeronautics

Go to...

- Beginner's Guide Home Page

*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *