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From: Timo Nieminen on 12 Apr 2010 00:14
On Sun, 11 Apr 2010, Ken S. Tucker wrote:

> Hi Timo,
> I reviewed this link, looks ok.
> http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)
[cut]
> I think the suggested fixes were impaired by a lack of understanding,
> scale, as you suggest, is part of it.
> The tourist attraction theory is novel :-).

I wasn't serious in that being a reason it wasn't fixed, but it was quite
a little local tourist attraction.

People learn. Look at what happened with the Millenium Bridge in London:
http://en.wikipedia.org/wiki/Millennium_Bridge_(London)

> > Ken: There's a Maxima-Minima problem in calculus, that involved
> > moving the exit orfice vertically to max the stream, (how far
> > out it would go squirt horizontally).
> > The column height was fixed.
> > Maybe I got spooked, but I found it tough.
>
> Timo: That makes it a hard problem for an introductory course. It's
> really
> three problems in one: Bernoulli, projectile motion, and the min-max
> problem. The difficulty of combined problems like this is non-linear;
> this is more than 3 times harder in total.
> The easy way is to think that the range is zero for y=0, and zero for
> y=h, and will be maximum somewhere in-between. Avoid square roots, and
> maximize range^2, rather than range. Range^2 = f(y,h) looks like it'll
> be quadratic in y, so df/dy will be linear, so maximum for y=h/2. No,
> I don't expect students to do this - just to grind through the maths.
>
> Ken: Yes you understand the problem. It's the parabolic
> trajectory of the fluid to obtain range that tweaked me.
> Certainly pressure and height can be assumed linear,
> (as well as a zero viscosity, incompressible fluid).
> You might be right, but you know math, I gotta prove it.

"Calculate the range" of a projectile is a standard exam question. 2D
vector motion, motion at constant velocity (horizontal), motion at
constant acceleration (vertical), it combines a lot of stuff in
introductory mechanics. Make just part of a bigger Bernoulli problem, and
ask for the optimisation, and it would be really nasty exam question. But
a good back-of-chapter question, for the same reasons.

> You know about impedance matching to transfer 'Power',
> such as W=V*I is matched with W source to a resistive
> load, fairly straighforward, but the water column problem
> has a ballistic trajectory complicating it.

Nice analogy, and interesting approach.

I wouldn't put your version of the Bernoulli bottle question on an exam -
the optimisation is too much math, not enough physics, I think, for the
type of course that basic Bernoulli problems and basic projectile problems
are appropriate for. I don't think it's a great exam question without
that, either.

I always did it for real, with a real bottle and real water. The
projectile part was just to convert a measurement of the range into a
value for the squirt-out speed. That's when the fun starts, since this is
always less than the Bernoulli-predicted speed, anywhere from about 1/2 to
maybe 5-10% too low, depending on the size of the hole. Since I did this
for real, I didn't think it worth repeating on the exam.

Here was a fun question I put on an exam once:

How large does a hot air balloon need to be? The density of
air is approximately 1 kg/m^3. Note any reasonable assumptions
you make if needed.

The last sentence is hand-holding, but sometimes you need to do some of
that. Students didn't need to do it, but they were welcome to try.

(Don't read s.p.r, added s.p.)

--
Timo
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