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THE SQUARE-CUBE LAW & SCALING FOR RC SAILPLANES

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by Michael Selig, © Copyright 1990 and 1998

This
past summer I, unfortunately, found myself at the controls of someone
else's large Sagitta XC (cross-country) after having been calibrated
for years to fly my 2M Prodigy. The sailplane was high and downwind
when I pushed the nose over to penetrate upwind. I had just reached
what seemed to be a moderate and safe speed when suddenly the wings
folded, and the glider tumbled to the ground in pieces – a fuselage
and two wings. The problem was that I attempted to fly the large XC at
speeds that appeared to be on the order of my small 2M. On another
occasion, a friend of mine scaled a 2M Prodigy up to XC size. His
initial reaction to the first few flights was that the 2M flew better
– it "seemed" to fly faster than the XC. As further examples, a scale
model sailplane looks faster than the full scale, a sparrow looks
faster than a crow, and a Learjet looks faster than a DC-10.
So that you will not be easily deceived by what you see, it is
worth going over why large aircraft appear to fly slower than small
aircraft.

If we consider a class of aircraft of similar geometry and
construction (for example, balsa and MonoKote or fiberglass and foam),
the weight will scale as the cube of a characteristic length. This
follows generally from mass being equal to density times
volume. Taking the wing span as the characteristic length, we can
therefore write:

Likewise for the wing area we have

These relations together form what is often called the
"square-cube law," from which a wealth of useful information may be
derived. For instance, these equations can be combined to give

Thus if a model is scaled up to twice its size, the wing loading will
double if the construction materials are similar. This relationship
can be used to estimate the weight of new designs based on old ones. It
is only necessary to determine the proportionality constant. Also,
from the equation for lift

we can obtain

This shows that if we increase the span and keep geometrically
similar models with the same type of construction, then the speed at a
given lift coefficient will increase with the square root of the wing
span. It does not yet show why large aircraft look slower than
small aircraft.

When an aircraft is flying against a blue sky, the only relative
measure of length or size is the aircraft itself. Therefore, we can
only base the speed on how long it takes the aircraft to cover some
characteristic distance relative to its own size. If we take the wing
span as this characteristic distance and call the time that
it takes the aircraft to travel one wing span the "characteristic
time," then we can write it as

From (4) we can than then write

Immediately we see that as the span increases so too does the time it
takes to travel a distance of one wing span.

Finally, we can compare the "speeds" of a 2M and XC sailplane by forming
the ratio

If the wing span of the larger plane (denoted as 1) is twice the smaller
(2), then the time it takes the larger plane to travel one wing span is
41% longer than the smaller. Although it is actually flying 41% faster
(see equation (5)), it looks 41% slower. That is a big difference.

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