•
Measure
L
, the unstretched length of the rubber band by placing the band on a flat surface and
put a ruler or pen on top so that it lies flat. Then use the travelling microscope to measure this
horizontal length of the band.
What do you find? Probably your data points lie on a gentle curve, not on a straight line at all —
if this isn’t obvious, hold the graph nearly level with your eye and look along the line of points.
With the increased precision of measurement, the elastic behaviour is clearly nonlinear.
Further investigation of the nonlinear behaviour
If the expression
F = Kx
is inappropriate, can we find a better description? Suppose the true
relation is a power series:
Note that we have divided the extension
x
by the original unstretched length
L
to give the
fractional
extension, often called the
strain
— this ensures that every power term of
x/L
is
dimensionless so the coefficients
A
,
B
,
C
, … all have the units of force. You have seen that the
linear ‘law’, taking just the first term of the series, is quite a good first approximation, so let us
investigate the effect of adding just the second (
quadratic
) term. Since
x
is much less than
L
this
term will be much less than the first, even if the coefficients
A
and
B
are similar. Your results
probably show that
F
increases slower than a straight line as the extension
x
increases, which we
can achieve by making the coefficient
B
negative. If we also try putting
A
numerically equal to
B
we have the simplest possible nonlinear expression:
•
Tabulate (
x/L
) – (
x/L
)
2
and plot this as the ordinate versus the mass
m
as abscissa. Draw your
own conclusions.
2–3
Laboratory exercise 2
Some final algebra
Even more precise investigations have shown that the expression:
is a good description of the elastic behaviour of rubber. Here
S
stands for the stretched length,
L
+
x
.
•
By expanding this expression as a power series in
x/L
try to
derive
a series in which the simple
quadratic expression you used above appears as the first two terms. You will need to use the
binomial theorem in your expansion; it is as follows:
where we must have
y
2
< 1 and
n
can be positive or negative.
•
How much more accurate would your measurements have to be if you wanted to show
experimentally that the next (cubic) term in the expansion was needed? Hint: roughly estimate
the size of this term for largest masses you have used.
3–1
Laboratory exercise 3
Laboratory Exercise 3 – NUCLEONIC MEASUREMENTS
Introduction
Experimental techniques in nuclear and elementary particle physics can be extremely complex,
requiring expensive apparatus and sophisticated treatment of data. However many of the
principles can be illustrated by measurements that use radioactive sources, fairly familiar
apparatus, and only simple arithmetic to derive results. This exercise uses a sample of the
radioactive isotope cobalt60 (
60
Co) and a Geiger counter to demonstrate the absorption of
!

rays; it also illustrates the treatment of errors due to the variability of repeated measurements.
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 Summer '09
 Electrical resistance, Electrical network, Voltage drop, Thermometer, Geiger tube