Jonathan Struebi Instructor Green MATH 106 March 15, 2004 Applying Hooke’s Law to Rubber Bands             Hooke’s law goes back to the 17th century when English physicist Robert Hooke discovered that elastic objects such as springs stretch in proportion to the force that acts on them. Up to this day this law holds true[1] and has become very important. Elastic objects that obey Hooke’s law are used everywhere in our lives. They are used in scales, watches and many other applications. Rubber bands, which may or may not obey Hooke’s law, have increasingly become important. They are used to secure packaging, as energy storage devices in cars and of course to play “cops.” Rubber bands have many advantages over springs. They are less expensive to produce, they weigh less and they can store more energy than springs. But can Hooke’s law be applied to rubber bands? There is much debate on this issue. While most agree that Hooke’s law does not apply to the entire range of motion of rubber bands, some argue that there is a range in each rubber band where Hooke’s law applies. This issue is important because, if Hooke’s law applies to rubber bands, it would expand their use to include measuring instruments such as watches and scales. In this project the argument that Hooke’s law applies to rubber bands is put to an experimental test and the results interpreted from a mathematical perspective.                         In order to test the hypothesis that Hooke’s Law applies to rubber bands two variables have to be measured accurately, force and distance. To do this the following setup is made. A clamp is attached to a vertical steel bar firmly screwed into the table, as can be found in a chemistry laboratory. A metric ruler is attached to the steel bar. An S-piece, available at most hardware stores, is hung over the clamp. It is this S-piece onto which the spring, respectively rubber band is hung. A second S-piece is now hung over the bottom of the spring. Carefully a piece of wire is attached to the second S-piece and bent so that it barely touches the ruler in the background, but not creating friction. A fleece jacket provides cushioning so that in the event of system failure the heavy weights don’t chip the table. Eye goggles protect from possible debris. Increasing weights are now attached to the system and the experimental phase begins (see “Experimental Setup.”)             At first an experiment is conducted using a spring with known spring constant. This experiment is conducted to test the experimental setup and method. The values obtained confirm the experimental setup. The measured spring constant is 0.0995 and the graph of distance vs. force (see “spring”) is a straight line. According to the manufacturer the spring constant is 0.1 and according to Hooke’s law F(x) = k*x, the distance is directly proportional to the force exerted on the spring, thus the graph of distance vs. force will produce a straight line, as seen in the experiment. With the experimental setup confirmed, the test is repeated three times using rubber bands increasing in size and strength. The first two tests using rubber bands are conducted using force increments of 0.196 Newton (0.020kg * 9.8 m/s^2) and the third test is conducted using force increments of 9.8 Newton. All these tests are started at force and distance from equilibrium equal to zero. The first test gives values that, if distance vs. time is graphed, produce the right half of the parabola y=k*x^2, where k is a constant (see “rubber band A”.) However, it should be noted that the graph in this case does not represent the entire range of motion of the rubber band. In the second and third experiment using rubber bands the rubber bands are stretched to their limit. The graphs obtained resemble graph “rubber band A” at first. Then as the rubber bands are stretched even more, it seems that there is a point of inflection and the graphs start flattening out. They become almost horizontal, shortly before the rubber band snaps (see “rubber band B” and “rubber band C”.) In the middle of the range of motion of the rubber bands there seems to be an area in which the slope is constant and thus Hooke’s law would apply. In order to test this mathematically a fourth degree polynomial fit is calculated using the regression capabilities of a graphing utility (see “calculations”) and the second derivative taken from both curves. The second derivative is then graphed. Because the graph of the second derivative does not touch, or closely approach, the x-axis it is concluded that Hooke’s law does not apply to rubber bands.             To expand the discussion beyond Hooke’s law the integral of the fourth degree polynomial fit is taken from the appropriate equations (see “calculations”). The integral is then evaluated within the bounds starting from equilibrium position and going to maximum extension. Maximum extension in case of the spring means stretching the spring as far as possible without changing its elastic properties. In the case of the rubber bands maximum extension, it means stretching the rubber bands until they snap, using the last values obtained before system failure. Comparing the values obtained from the integral calculations it’s evident that rubber bands store significantly more energy at maximum extension than springs of comparable mass and size. Specifically dividing the energy stored in rubber band C at maximum extension by the energy stored in the spring at maximum extension (see “calculations”) one obtains the number 65,000. This means that in this experiment, 65,000 times more energy could be stored in the rubber band when compared to the spring of similar mass. While this is impressive, the immense energy rubber bands can absorb will not be further discussed in this paper. This investigation’s prime purpose is to see if Hooke’s law applies to rubber bands. The answer is no. Further investigation of the ability of rubber bands to store energy however is strongly recommended for future research.               While rubber bands are useful in our daily life, Hooke’s law does not apply to them. This is probably the chief reason why rubber bands are rarely used in instruments of measurement, an area in which springs clearly prevail. However, reviewing the experiment and calculations it is evident that rubber bands will continue to have wide applications. The reason being that they can absorb a fascinating amount of energy. With this in mind there is no doubt that rubber bands will continue to be indispensable in our life.               Calculations: (Completed using TI 89)   Spring: Linear Regression: y = .09948x + .006271 corr. = .999862 R2 = .999724 Integral with respect to x of y from 0 to .892 (entire range of motion[2]) = .045 Ncm = .00045 Nm     Rubber band B: Quart Regression: y = .009881x^4 – .167417x^3 +.8346426x^2 + .615986x – .077686 R2 = .99964 Dy2/dx = .118575x^2 – 1.0045x + 1.66925 (Parabola with vertex 4.2, -.45) Integral with respect to x of y from 0 to 7.25 (entire range of motion) = 45.6 Ncm = .0456Nm   Rubber band C: Quart Regression: y = 9.279537E-8x^4 – 1.202637E-5x^3 – .002699 x^2 + .734843x –. 588141 R2 = .998637 Dy2/dx = .0000011135x^2 – .0000721582x – .0053974259 (Parabola with vertex 31, -.0066) Integral with respect to x of y from 0 to 107.8 (entire range of motion) = 2943 Ncm = 29.4 Nm     Comparison of energy in rubber band vs. energy in spring of comparable weight: 29.4 Nm (rubber band C) / .00045Nm (spring) = 65300 times more energy in rubber band.   [1] In reality Hooke’s law does not perfectly model the behavior of springs. For the purpose of this paper however it is assumed that Hooke’s law holds true for springs. [2] Entire range of motion means that the object is extended as far as possible without changing its elastic properties.