x^{5} might seem like a pretty arcane and unfathomable beastie, but
it has got some perfectly down to earth applications. We even find a use for it down
here in the club basement, which is about the most unsophisticated place I can
think of. It's to do with the big fan that pumps fresh air through the ventilation
system to all parts of the club. Now if you enlarge one of these fans by a certain
scale factor x then what sort of an increase in your electricity bill would
you expect? Well knock me down with a feather and call me Nancy if it isn't a factor
of x^{5}.
That's right, a whopping x * x * x * x * x.
This is assuming that you keep the basic shape of the fan the same and that you
don't alter the speed of rotation.
This means that if you simply doubled the size of the fan then the power required
to drive it would go up by a factor of 32. And if you were to triple
the size of your fan it would increase by a factor of 243!! (That's because 3^{5}
equals 3 * 3 * 3 * 3 * 3 which equals 243).
If that's not astounding then I don't know what is.
It's difficult to come up with a clear explanation of this enormous increase in
power requirement, but try this one:
For a start, when we scale up the size of the fan we're making it x times
taller and x times wider and x times deeper. So the volume of air
held in the fan is going to go up by a factor of x cubed.
So I can probably convince you that x cubed times as much air will get
flung out of the fan on every revolution and therefore the power required will
go up by at least x cubed.
But there's another completely separate effect of enlarging the fan. This effect
is caused by the fact that the tips of the fan blades are now x times further
away from the centre of the fan. That means that every litre of air that leaves the
fan is going x times as fast as in the original fan. And that means that
the kinetic energy in each litre of air whizzing out is x squared
times what it was before (see discussion of x squared).
We end up with x squared times as much energy in x cubed times as
much air so we have to put x^{3} * x^{2} =
x * x * x * x * x = x^{5} times as
much expensive energy in every second.
It makes no difference if we slow down the leaving air to the same speed as before.
If we did that then every litre of outgoing air would get squashed up against the slower
moving air in front of it and it's kinetic energy would get converted into pressure
energy. So we'd have x cubed times the volume or air at x squared times
the pressure which still needs x^{5} times as much
power. Actually, some of that energy would probably end up as heat but we still can't
win.
