An airplane in flight can be maneuvered by the pilot using the aerodynamic control surfaces, the elevator, rudder, or ailerons. As the control surfaces change the amount of force that each surface generates, the aircraft will rotate about a point called the center of gravity. The center of gravity is the average location of the weight of the aircraft. The mass (and weight) is actually distributed throughout the airplane, and for some problems it is important to know the distribution. But for total aircraft maneuvering, we need to be concerned with only the total weight and the location of the center of gravity.

How do engineers determine the location of the center of gravity for an airplane which they are designing?

In general, determining the center of gravity is a complicated
procedure requiring the use of calculus as discussed on another
page.
The figure shown above is a
simplified version that can be used by secondary students. For this
figure, we assume that we already know the
weight and location, relative to some reference location, of each
of the major components of the airplane (the fuselage, vertical and horizontal
tails, wings, engines, fuel, payload, etc.). The total weight of the
aircraft is simply the sum of all the
individual weights of the components. Since the center of gravity is
an average location of the weight, we can say that the weight of the
entire aircraft times the location of the center of gravity is equal
to the sum of the weight of each component times the distance of that
component from the reference location. (**To simplify:** the
center of gravity is the mass-weighted average of the component
locations.)

For each of the components, the **center of gravity** is the
average distance of the weight from some reference line. If we have a
solid block of uniform material, the calculation is simply the average of
physical dimensions. (If we have a rectangular block, 50 X 20 X 10,
the center of gravity is at the point (25,10, 5) ). But if the block
is nonuniform in either its density or its shape, we have to resort
to more sophisticated averaging techniques. We can generalize the
technique discussed above.
If we had a total of
"n" discrete components, the center of gravity (cg) of the aircraft times
the weight (W) of the aircraft would be
the sum of the individual (i) component weight times the distance from
the reference line (wd) with the index i going from
1 to n. Mathematicians use the greek letter sigma to denote
this addition. (Sigma is a zig-zag symbol with the index designation being
placed below the bottom bar, the total number of additions placed over
the top bar, and the variable to be summed placed to the right of the sigma
with each component designated by the index.)
So this equation says that
the center of gravity times the sum of "n" parts' weight is equal to
the sum of "n" parts' weight times their distance. The discrete
equation works for "n" discrete parts.

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