Re: time squared & time cubed

["John S. Denker" <jsd@MONMOUTH.COM>]

At 12:50 PM 1/16/02 -0400, Tim O'Donnell wrote:
> > 1) I understand time squared and time cubed.  I know what
> > that means.
>
> Help explain it to me.  I can do problems with those
> units, but they do not convey a meaning to me.
> > 2) I don't understand left/right or up/down in time.  And if
> > that did mean anything, it probably wouldn't have anything
> > to do with time squared or time cubed.
>
> With meter^3 i can go forward/backward, left/right and
> up/down. Time I can go forward/(think about going
> backward) , but what happens after that.
>
> > Was there a reason for hypothesizing a connection between
> > idea (1) and idea (2)?  I'm not seeing it.
>
> Just to make an anology the regular student and me can
> visualize.

Wow, I was totally not understanding the question.
I think I've got a clue now.  Try this:

1) IF (big if) you have a cube and scale it up in
all directions by a factor of X, its volume goes
up by a factor of X to the third power, which we
commonly call X cubed.

2) That scaling law isn't restricted to cubes.  It
applies to everything that can be scaled up in three
dimensions.  This includes parallelepipeds, tetrahedrons,
dodecahedrons, and oddly-shaped things like potatoes.
If you scale them up in all directions, the volume
goes up like X to the third power.  So we see that
calling it X cubed is suggestive, but may or may not
suggest the right things.

3) The converse does !!not!! hold.  If you have something
that depends on something else to the third power, there
is !!no!! reason to expect that this is due to three factors
of the form length*width*height.  For example, consider the
flexure of a springy plank, such as a high-diving board.  If
you make the board thicker by a factor of X, leaving the
other dimensions the same, its stiffness goes up by a factor
of X to the third power.  Calling it X cubed is a bit of a
misnomer;  there is no cube of size X involved.  Yes, there
is multiplication going on, but it is not length*width*height
or anything like that.  (There is something of size X that is
multiplied by a mechanical advantage of X, roughly speaking.)

Similarly, in electronics, I^2 R heating is pronounced "I
squared R" but it does not involve a square of size I on a
side.  Perhaps we should have called it "I to the second
power R" to avoid the misleadingly suggestive term "squared",
but I'm afraid it's too late to lock that barn;  the horse
escaped eons ago.

For homework:  Collect as many examples of you can:
 -- Physical laws involving a second power having nothing
    to do with area.
 -- Physical laws involving a third power having nothing
    to do with volume.

For "extra credit":  Collect physical laws involving
fractional powers.  For starters, how does the magnetization
of a piece of iron depend on (T-Tc) as the temperature
T approaches the Curie point Tc?
  http://www.nobel.se/physics/laureates/1982/press.html

Bottom line:  Don't think of "X cubed" as a cubical object.
Think of it as X multiplied by X multiplied by X.  The
multiplication operator is a big-time party animal;  it can
multiply all sorts of things, not just length*width*height.