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

1

where is the angle between the vectors, and . For a finite displacement along a path, the work is the sum of the infinitesimal amounts of work:

2(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;

3

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

4(2)

where the quantity 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 one-dimensional 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, , (2) may be integrated to yield the 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.)

1 Figure 1. In one dimension, the work is the area under the force vs. displacement curve.

Apparatus

o rubber band (width: 1/8 to 1/4 inch; length: 3 to 3 1/2 inches)

o table clamp and uprighto 1 50-gram slotted weight

o meter sticko 1 100-gram slotted weight

o 2 small rubber bandso 4 200-gram slotted weights

o pendulum clampo 2 500-gram slotted weights

o long stem 50-gram weight hangero 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.

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.

force
(9.81 x 10-3 N) / Stretching / Contracting
meter stick
reading (10-2 m) / length of band
(10-2 m) / meter stick
reading (10-2 m) / length of band
(10-2 m)
0 / so / 0 / sf / sf - so
200 / s1 / s1 - so
400 / s2 / s2 - so
600
2000 / smax / smax - so / smax / smax - so

Figure 2. Suggested form of the data table.

8)When the maximum mass of 2000 grams is reached and the meter stick reading, , is recorded, reverse the entire procedure. That is, remove a 200-gram weight, wait 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, , 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

5

4)Trapezoidal rule. (Refer to Appendix IV.) On the curve, choose points of equally-spaced 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 equally-spaced 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 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'.

2Figure 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?