Here's a nice example of x^{4} in use. x^{4} simply
means x * x * x * x and it's pronounced
"x to the power 4" or even just "x to the four". It doesn't get a special
name for itself unlike x squared or x cubed. Anyway, the problem is these
I beams holding up our main dancefloor. They're allowing the floor to sag in the middle
when lots of people are dancing on a busy night.
Now not many people know this, but as you scale up the cross section of a beam by
a certain factor x then it's stiffness goes up by a factor of x^{4}.
Or, to look at it another way, the sag in the middle of the beam gets smaller
by a factor of x^{4}. So if you double the section of a beam (ie make it
twice as wide and twice as high) then it will get 16 times stiffer,
because 2^{4} = 16. In other words it would only sag 1/_{16}^{th}
as much under a certain load; or it would take 16 times as much weight before it would droop
the same amount. This is assuming that the bigger beam is spanning the same gap and still
made of the same kind of metal.
Now this is a quite astounding increase in stiffness. It means that even if you only
increase the cross section of the beam by 20% (ie a factor of 1.2) then the beam
will get stiffer by a factor of 1.2 * 1.2 * 1.2 * 1.2 = 2.0736.
So you've improved the stiffness more than twofold just by scaling up by 20%.
It's not easy to see why this is true, but try this explanation: Imagine the
beam as being made up of strands or fibres of material running end to end like dry
spaghetti. Now when
you put heavy load in the middle of the beam the fibres at the bottom get stretched
and the fibres at the top get compressed. And these fibres act like springs resisting
the stretch and trying to straighten the beam again. Well when you scale up the beam
cross section by some factor x it's like adding extra fibres. You're adding
extra fibres at the sides and adding extra fibres top and bottom.
The sheer amount of metal in the beam goes up by a factor of x squared.
So it shouldn't be difficult to convince you that the beam gets stiffer by
at least a factor of x squared.
But the average fibre in the bigger beam is also twice as far out from the middle of
the beam. and this gives you two separate additional advantages in
terms of beam stiffness. Take a single fibre right at the bottom for example.
In the scaled up beam it's now x times further away from the centre line.
Now let's imagine that the beam is hinged in the middle and that this one fibre is
doing all the work of opposing the sag. If this fibre gets stretched by 1cm then
it will generate a certain opposing force (tension) trying to unstretch itself
again, just like a spring. It would have done exactly the same thing in the smaller
beam. But because the fibre is now further away from the middle it will produce a
stronger resistance to the bending of the hinge. It's just
like putting longer handles on a pair of pliers; although the tension in the fibre
is the same it exerts a bigger effect trying to straighten the beam, because it's
got better leverage just by being further away for the hinge.
That's our first advantage from the fibre being further from the
centre line. We have been comparing a fibre stretched 1cm in the scaledup beam with
a fibre stretched 1cm in the original beam.
But in reality, for a certain depth of sag, the bottommost
fibre in the bigger beam will get stretched x times further than before!
This is the second independent effect of the fibre being
further out from the centre line. So if the fibre was stretched 1cm in the original
beam then it'll be stretched 2cm when we scale up by factor of x=2.
And because the fibre is acting as a spring then the more you stretch it the harder
it'll pull back. The fibre now has better leverage and it's
pulling harder than before.
So there you have it. You've ended up with a beam with x^{2} times
as many fibres in it, and each of those fibres is pulling x times as hard
and with x times better leverage for counteracting the sag of the beam.
So the net result is a beam that's x * x * x * x
times stiffer. And that's x^{4}!
