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8 INTRODUCTORY PHYSICS ACTIVITY: KNOTS
L. Hiller , North Tonawanda.
This exercise is intended to be used early in the year to develop the ability in students to not only create graphs of data, but to interpret the meaning of the results. The activity takes two 40 minute periods to complete.
THE PROBLEM:
Our engineering firm has been hired to construct a suspension bridge. The
Bridge will be held up by hundreds of 50 cm thick cables. To secure the roadbed to the scaffolding of the bridge, 4 knots must be tied in each cable.
The length of the cables AFTER the knots have been tied, must be exactly
60 meters. The cable is very expensive and we dont want to make any more than we actually need. In what length must these cables be originally created in so that they will have the desired length after installation?
WORKING OUT THE PROBLEM:
We dont have any 50 cm thick rope, and even if we did, how could we possibly tie knots in it??? What are our options?
What if we sc ale down the problem and look at how knots behave in more manageable strings?
Each pair of students is given a 1 meter long piece of cord, spanning a range of diameters. (My class set cost a total of about $12 at Home Depot.)
The students offer immediately that the rope gets shorter as you tie knots in it but are unsure if the length change happens in a predictable way.
The students are asked to take data on their rope and create a graph.
Care must be taken to avoid double knotting. Students with thinner ropes
should a few knots in between measurements.
The graphs will look like this: INDIVIDUAL DATA
100
Length
y = mx + b (cm)
L = -mK + Li
# of Knots
Each rope will yield a nice straight line. Have the students create the
equation of the line by answering these questions:
What is the y intercept? What does it MEAN?
What is the slope of the line? What does it MEAN?
EMPHASIZE THE UNITS!!!
Of course, the studen ts see that there is a pattern in their own data, but is
their a pattern in the classes data that will allow us to solve our problem?
Each group will get a different slope value.
Ask the students: Why does your rope lose 2 cm every time you tie a knot and their rope lose 8 cm every time they tie a knot?
The students will decide quickly that the thickness of the rope controls how
fast the length changes.
Give the kids access to calipers and create a new pooled data table with
the thicknesses of each rope and the slopes from the first graph. Use this
pooled data to have the students create a new graph.
These graphs will look like this:
POOLED DATA
y = mx + b |m| (cm/k)
|m| = MT + 0
|m| = MT
thickness (cm)
This gra ph shows us that their is a definite relationship between the thickness and how quickly the rope changes length. The slope of this graph
should come out around 10 (cm/k)/cm. Again, emphasize the units!
We are now fully equipped to solve our problem:
|m| = (10 (cm/k)/cm)(50 cm) = 500 cm/k
L = -mK + Li
6000 cm = -(500cm/k)(4k) + Li
Li = 8000 cm = 80 m.
A GEOMETRIC APPROACH TO THE PROBLEM:
Where does the lost length go?
When you tie a knot in a rope, you are just wrapping the knot around itself.
Thats why the thickness matters. The lost length ends up surrounding the
original chord.
Cross-section
of a knot
BLOW UP OF CROSS-SECTION:
The chord has a thickness, T. The circle
formed by the knot has a total diameter of
3T.
The missing length of the chord is the
circumference of this circle.
C = D
C = (3T)
C = 3(T)
This 3 is the 10ish slope we got from the algebraic solution!
C = 3(50cm) = 471 cm, which is lost for each knot tied.
4knots(4.71 m) = 18.84 m + 60 m = 78.84 m
% difference = ((80-78.84)m 80m )x100% = 1.45%. Not bad.
Value of the Lesson:
* Non-threatening, yet interesting subject
* Hands on problem in the first few days of school
* Team building exercise in cooperative data taking
* Powerful ideas of Scaling and Extrapolation
* Transparency of equivalence of algebraic and geometric techniques
* Meaningful discussions of averaging data with best-fit line
* Build students confidence in physical interpretation of algebra early Z N DSET 2 H + P ] 9
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