Togetherness "Center of Gravity"

[Dave Monroe]

"The next thing is to decide where to put what in the
cargo spaces. To have a safe flight, the center of
gravity of the plane must stay between certain body
stations. Almost always there is extra freight, like
batteries and test sets, to be sent along with
missiles and airfoils...." (Togetherness)


"the center of gravity"

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 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. 

http://www.grc.nasa.gov/WWW/K-12/airplane/cg.html

And see as well ...

http://www.grc.nasa.gov/WWW/K-12/airplane/forces.html

Recall, e.g., ...

"In the static space of the architect, double integral
now and then, early in his career, to find volumes
under surfaces whose equations were known-masses, mo-
ments, centers of gravity. But it's been years since
he's had to do with anything that basic." (GR, p. 301)

"The double integral stood in Etzel Olsch's
subconscious for the method of finding hidden centers,
inertias unknown, as if monoliths had been left for
him...." (GR, p. 302)

"Perhaps, because the Rocket coming down was lighter
by lo tons of fuel and oxidizer, the shift in the
center of gravity was making it unstable." (GR, p.
424)

"We need not dwell here on the Primary Problem [...]
except to emphasize to to beginning students who may
be prone to Schwarmerei, that terms referring to the
Subimipolexity such as 'Core' and 'Center of Internal
Energy' possess, outside the theoretical, no more ...
than do terms such as 'Supersonic Region' or 'Center
of Gravity' in other areas of Science) ..." (GR, p. 700)


		
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