I'd like the list's opinion and suggestions to address the luncheon
meat issue for a freshman-level recitation problem. The class has
finished kinematics and is now working on N2 (summing forces and
finding accelerations).
The problem has been used in recitation before so it's not original -
I'm sure some of you have seen it. I found myself criticizing the
original problem for reasons stated here - too cooky booky, with
essentially all of the required information given. Rather than create
a new problem (I am notoriously uncreative), I have attempted to
"fix" this one - please note that I'm not aiming at the level of the
Maryland problems just pointed out, which I agree are good. My newer
wording goes as follows (discussion further below):
"You are a props engineer at the Playmakers Theatre, helping with a
children's production of Peter Pan. At a certain point in the play,
the script calls for Peter to "fly" down from above. In the past this
has been accomplished by attaching the performer playing Peter to a
wire of some sort, which runs over an overhead pulley to a
counterweight supported by an inclined smooth surface in the shape of
a right-angled wedge. The director tells you that Peter must be
lowered from a platform at the top of the wedge, to the floor. In
order for the stunt to fit in with other parts of the play, the
lowering must be accomplished in approximately 2 seconds (3 sec will
be too long, 1 sec will be too short). 'I want it to look like
free-fall, only slower. Somewhere around here are some wires,' says
the director. 'Whatever you find, make sure it is the thinnest
possible wire so that the audience can't see it.' You dig around and
find 35ft of piano wire and also a wire that has a worn label on it:
'.125in wire, tensile strength 2000psi.'
Remembering your freshman physics lessons, you use a caliper to
measure the piano wire in several places and find that it is quite
uniform, with a nominal diameter of 0.015in. At the same time, you
confirm that the steel wire is indeed 1/8inch in diameter. With a
tape measure, you determine the dimensions of the wedge incline: 10ft
high at its vertex, 10ft wide, extending 9ft deep along the floor.
You also measure the counterweight to be 50 pounds, and by letting it
slide down the incline a few times, you see intuitively that friction
will not be a factor. Assuming that the strength of the wire is the
limiting factor, rather than any harness or connectors used, report
to the director whether you can safely rig the stunt with the
materials at hand, along with any additional observations."
===============
I debated not including the wire thicknesses but did because piano
wire comes in many thicknesses, as is true for other wire. I also
didn't want to make it an algebra problem with all variables
unassigned. IOW, I wanted to specify things to the degree they might
be found in real life.
The director's wishes suggest students reasonably assert that the
problem is constant acceleration ("like free-fall") but less than g.
The time requirement causes the students to have to bracket their
answers using reasonable assumptions (for example, 1.5s < t < 2.5s).
In analyzing the dynamics of the problem, the following limits can be
found (mg = weight of Peter, a = acceleration, T = tension, t = time):
mg = 47lb, a = .33g, t = 2.5s, T = 42lb
mg = 71lb, a = .91g, t = 1.5s, T = 51lb
In other words, the "additional observations" should include that the
performer playing Peter should reside in a certain weight range.
Luckily, this range is rather elfin, but there is no exact answer -
the director will be able to choose from variations.
Next though, the students must address the wire situation - they will
have to address the concept of tensile strength, which, while outside
the scope, they should be able to briefly learn concept-wise, and see
intuitively how it applies. There is also discussion to be had about
yield strengths vs ultimate strengths etc, but while very real-world,
I consider these to all be "second order" for this level of problem.
My searching around (wiki article on ultimate strength for example)
suggested ~ 2000MPa for the piano wire. For the other wire, students
are given the strength, but in a way that is not unreasonable in real
life (a tag on the wire). Given the diameters, I calculate breaking
strengths on the order of 51lb for the piano wire, and 25lb for the
other wire. Thus, to work pretty much as described with some kind of
safety factor (cue that discussion among the students), they will
need more than 1 strand. To get more than 100lb strength, they will
need 2+ strands of piano wire, but 4+ of the other wire. The students
may find this counterintuitive (good), but the piano wire is clearly
superior for the desired visibility. For the wedge though, to
reasonably rig the stunt, the 30ft of available piano wire may be
used up, so there are still choices to be made.
So, I'm looking for comments and suggestions - how to improve the
problem without discarding it from consideration, wording or
content-wise, and/or how to make it more realistic, more challenging,
but not too open-ended. Also, hopefully I haven't done the analysis
wrong! One criticism is that 2000psi strength for the .125 wire
appears to be rather low, but at least we don't know what kind of
wire it is.