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
about a point called the
center of gravity.
The center of gravity is
the average location of the weight of the
aircraft. The 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
How do engineers determine the location of the center of
gravity for an airplane which they are designing?
An airplane is a combination of many parts; the
plus the payload and the fuel.
Each part has a weight associated with it which the engineer can
estimate, or calculate, using Newton's
w = m * g
where w is the weight, m is the mass, and
g is the gravitational constant which is 32.2 ft/square
sec in English units and 9.8 meters/square sec in metric units.
To determine the center of gravity
we choose a reference location,
or reference line. The cg is determined relative to this reference
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 W times the location cg of the center of gravity
to the sum of the weight w of each component times the distance d
of that component from the reference location:
W * cg = [w * d](fuselage) + [w * d](wing) + [w * d](engines) + ...
The center of gravity is the mass-weighted average of the component
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 d
from the reference line (w * d) 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.)
W * cg = SUM(i=1 to i=n) [w * d]i
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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