This is suggested as an in-class activity on page 10 of _Merrill Physics_
by Zitzewitz, Davids & Neff.
Perhaps this group would enjoy discussing the pros and cons of this activity.
The objectives are clear (height and stability), the constraints are easily
stated (one sheet of paper, 30cm of tape, and 45 minutes or so of construction
time), and the judging is straightforward.
Factors that make it interesting include
-- The strategy and tactics of solving the problem are not specified.
-- There is more than one "right answer", i.e. different strategies
may yield comparable results.
-- Original and creative thinking yields tangible rewards.
-- A little preliminary experimentation yields tangible rewards.
-- It is truly a hands-on activity.
-- The fact that it is possible for something 2 meters tall and (mostly)
less than .5 cm in diameter to be stable is fairly impressive ... and the
fact that it can be made out of only one 8.5x11 sheet of paper and a bit
of tape is even more impressive.
I've had people say to me, "That's nice, but is it physics?" My smart-alek
answer is that is sure isn't chemistry or biology or math, and there aren't
any engineering classes in high school, so by default it must be physics.
A better answer is that if something is worth doing I don't worry about
whether it meets somebody's narrow definition of this-or-that discipline.
A third answer is that yes it is physics, if you lay the appropriate
foundation and pursue the appropriate follow-up.
Zitzewitz et al. follow up by asking the students to explain "What
architectural elements have been incorporated into your design?" ...
which I find bizarre. Not many HS students know what an "architectural
element" is.
The value of the exercise is greatly enhanced if the students can take away
some understanding of the _principles_ involved. The usual formulations of
this activity greatly shortchange the principles. Hands-on is great, and
"student inquiry" is great as far as it goes, but it doesn't go far enough,
not nearly far enough. The students are very unlikely to discover the
principles on their own ... I have to _teach_ the principles.
The principle of /stability/ is very important here ... but this activity
is *not* suitable for *introducing* the idea of stability. There are lots
of things the teacher ought to do to before starting this activity, to
make sure the students know about stability. At the very least, there is
the activity of making a cantilever of books (or wooden blocks) at the
edge of a table, continuing it far enough that the top block does not
overlap the table at all. I find it disappointing that Zitzewitz et al.
do not say anything about establishing the conceptual foundations for the
paper tower exercise.
Relevant principles include:
*) Stability: How big does the base have to be, to ensure stability?
-- The concept of center of mass.
-- The concept of stability: this requires the CM to be above the base.
-- The concept of _margin_ of stability: how _far_ can the CM move and
still be above the base.
-- Stability in terms of energy: how much energy does it take to knock
the thing over?
-- The connection to the idea of leverage.
*) The importance of symmetry. Any deviation from symmetry is going to
hurt the tower. An appreciation for symmetry is an essential item in
any good physicist's bag of tricks.
*) The importance of good tools. If you don't have the tool you need,
make the tool you need. I found it difficult to roll the paper as
required, and virtually impossible to tape it properly, without the
use of a mandrel. I found a pen that was the perfect size to serve
as my mandrel.
*) The concept of tradeoff. Some tradeoffs here are actually tricky.
For example, putting too much material into the base causes only a
proportionate loss of height, while putting too little material into
the base might cause a _disproproportionate_ loss of stability.
*) As mentioned above, experimenting with a "pilot project" is helpful.
*) The idea of scaling laws. Paper is remarkably resistant to shear in
the plane of the paper. The only reason it is flexible is that it is
very thin. Stiffness scales like the third power of the thickness,
which is a pretty strong dependence. The stiffness of a thin-walled
tube scales like the square of the diameter (at constant wall thickness)
if you can keep it from buckling. A paper tower 2m tall and .43 cm in
diameter has no problems with strength or stiffness ... its limiting
factor is stability, which depends only on precision, symmetry, and
balance. Scaling laws are a super-important part of physics, and have
been since the time of Galileo.
*) I found it helpful to make N-1 of the joints as rigid and secure as
possible, while leaving one joint (halfway up) adjustable. (I don't
know what to call the principle here, but if I hadn't done this, I
doubt I could have gotten my tower to be stable.)