Experiment 7
A Nonconservative Force
A rubber band is suspended from an upright. Starting from its unstretched length, the band
is slowly stretched until it extends a distance which is large compared to the unstretched length.
The force is supplied by adding successive amounts of weight to the end of the band. The band is
then allowed to slowly contract until the unstretched length is reached. This is accomplished by
removing successive amounts of weight. The work performed in stretching is determined, as well
as the work performed in contracting. The net work is then found.
Theory
The infinitesimal amount of work done by a force acting over an infinitesimal displacement
is defined as




d W = F  d r = | F || d r | cos  ,


where  is the angle between the vectors, F and d r . For a finite displacement along a path, the
work is the sum of the infinitesimal amounts of work:
 

W =  rr12 F  dr ,
(1)
where the integral is performed over the path. The force is defined to be conservative if the total
work done around any closed path is zero;


 F  d r =0 ,
for a conservative force. The usefulness of this property is that a potential energy function can be
associated with a force if and only if the force is conservative.
In one dimension, (1) reduces to
W =  xx12 F dx ,
(2)
where the quantity F dx can be negative or positive, according to whether the force lies along the
direction of displacement or is opposed to it. A necessary and sufficient condition for a onedimensional force to be conservative is that the force be a function only of the distance; in
particular, that it not be dependent upon the history of displacement.
If the functional form of the force is known, F = F(x) , (2) may be integrated to yield the
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work. If such a relationship is not known, or if the relationship is very complicated, the work may
be determined graphically as the area under the force versus displacement curve. (Refer to Figure
1.)
Figure 1. In one dimension, the work is the area under the force vs. displacement curve.
Apparatus
o
o
o
o
o
o
rubber band (width: 1/8 to 1/4 inch; length: 3 to 3 1/2 inches)
table clamp and upright
o 1 50-gram slotted weight
meter stick
o 1 100-gram slotted weight
2 small rubber bands
o 4 200-gram slotted weights
pendulum clamp
o 2 500-gram slotted weights
long stem 50-gram weight hanger
o plastic triangle or file card
The pendulum clamp is attached to the top of the upright. The meter stick lies along the
upright and is held in place by the small rubber bands.
Procedure
1) The rubber band should be new. Inspect it. If there are any tears, then replace it.
2) Set up the apparatus according to the instructions in the Apparatus section. Loop the
rubber band over the pendulum clamp.
3) Prestretch the rubber band by suspending 2000 grams from the band. Wait roughly five
minutes, then remove the 2000 grams. Wait roughly five more minutes without any
weight on the rubber band.
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4) Once the experiment begins, the meter stick must not be moved. Record the position of
the lower end of the rubber band. Hold the triangle or the card against the meter stick in
order to determine the position.
5) Place 150 grams on the weight hanger for a total mass of 200 grams. Hook the weight
hanger to the bottom of the band and slowly allow the band to support the weight. The
stretching must occur monotonically; that is, the length of the band must never decrease
while the weight hanger is stretching the band. Wait approximately one minute then
record the meter stick reading of the lower end of the band. (This is necessary because
the rubber band continues to stretch for a period of time after the mass is added.)
6) Hold the hanger and slowly add a 200-gram weight. Allow the band to gradually
support this added weight. Remember to allow the rubber band to only lengthen. Wait
about a minute and record the position of the lower end of the band.
7) Continue to add 200 grams weights in the same fashion as above. When four 200-gram
weights are being replace by two 500-grams weights, it is important to hold the weight
hanger securely so that the band is neither stretches nor contracts during the transfer.
Once the transfer has been accomplished, slowly allow the band to support the weight.
Stretching
Contracting
force
(9.81 x 10-3 N)
meter stick
reading (10-2 m)
length of band
(10-2 m)
0
so
0
200
s1
s1 - so
400
s2
s2 - so


2000
smax
meter stick
reading
-2
(10 m)
length of band
(10-2 m)
sf
sf - so



smax - so
smax
smax - so
600
Figure 2. Suggested form of the data table.
8) When the maximum mass of 2000 grams is reached and the meter stick reading, smax ,
is recorded, reverse the entire procedure. That is, remove a 200-gram weight, wait
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about a minute, and then record the meter stick reading. Again, it is important that the
contractions occur slowly and monotonically. (The length of the rubber band must
never increase now.) The final meter stick reading, s f , corresponds to zero mass. The
suggested form of the data table is shown in Figure 2.
Analysis
On a single graph, plot force on the ordinate and length on the abscissa. For convenience,
force should be plotted in units of 9.81 x 10-3 N (as in the suggested data table) so that the value
of mass in grams can be plotted directly. The intersection of the axes should correspond to zero
force and length. For accuracy, it is important that the plot fills as much of the graph paper as
possible.
Draw a smooth curve through or near the points that correspond to increasing length
(stretching) and a smooth curve through or near the points that correspond to decreasing length
(contracting). Use arrows to indicate the direction of each curve, i.e., whether it corresponds to
stretching or contracting.
Find the work done in stretching and contracting by one of following methods:
1) Counting squares. Each square on the graph corresponds to the same amount of
work. By counting the squares under the curve and multiplying by this value, the
total work can be found. (There is some freedom in the choice of the size of the
square. It need not be the smallest square, but of such a size to provide the degree of
accuracy desired.)
2) Weighing. A photocopy of the graph may be cut along the boundary. Also a very
large rectangle corresponding to a known amount of work may be cut from the same
photocopy. Using a sensitive balance, the work corresponding to the area under the
curve can then be found by means of a proportion between mass and area.
3) Trapezoidal rule. Construct trapezoids from the data points. The area of each
trapezoids is
W(area) = 12 ( Fi + Fi + 1)(si + 1 - si ) .
4) Trapezoidal rule. (Refer to Appendix IV.) On the curve, choose points of equallyspaced abscissa values. The trapezoidal rule can then directly yield the work.
5) Simpson's rule. (Refer to Appendix IV.) On the curve, choose points of equallyspaced abscissa values that give an even number of intervals. Simpson's rule can then
directly yield the work.
6) Simpson's rule: alternative method. (Refer to Appendix IV.) The data for the curve
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can be used directly, provided they do not significantly deviate from the curve. The
data, however, has equally-spaced ordinate values, so the complement, W', of the
work is determined by Simpson's rule. (Refer to Figure 3.) The work, A,
corresponding to the rectangular area is easily determined. W is then found by
subtraction: W = A - W'.
Figure 3. Simpson's rule yields the area W' when the data is used.
Using one of the above methods or one of equivalent accuracy, determine both the work
performed in stretching and the work performed in contracting. State the method used and show
all relevant calculations. Express the values in joules. If, as recommended, units of 9.81 x 10-3
N and 10-2 m are used for the force and the distance, be sure that these factors have been included
in the calculation. (These actually need be inserted only at the end of the calculation after the
area has been determined.) From the two values of work, find the net work performed over the
entire path.
Report the values of the work done stretching, the work done contracting, and the net
work done in a results table.
Questions
1) Give a qualitative sketch of force vs. length for the case where the hanging mass is
nearly sufficient to break the rubber band. Clearly display the behavior in the region
that is near the breaking point of the rubber band.
2) Is the force exerted by the rubber band a conservative force? Explain.
3) The entire curve is referred to as a hysteresis curve for the rubber band. The net work
is the area inside this curve. Why?
4) What microscopic processes do you think are responsible for the nonconservative
force exerted by rubber?
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