P-3 PENDULUMInstructor's notes VALUE OF THIS EXPERIMENT The pedagogical purpose of this experiment is to illustrate how one investigates a natural phenomenon, isolates variables, and expresses the results with a mathematical equation (model). It is best scheduled early, before pendulums are discussed in lecture or text. The graph on log-log paper should be quite straight. A linear regression of student data gives g to within 3% of the accepted value. This is not a shoddy experiment, and the results are good if the student takes reasonable care. GRADING CHECKLIST Results.
Value of constant n in T = KL^{n} determined from graph. Should be near 0.5. Value of constant K = 2π/√g, determined from graph. Experimental determination of g. Error limits for each of the above, with supporting experimental support for each value. Graph.
Axis labels: quantity, symbol, units. Descriptive title (not "T vs. L"). Genuine and appropriate graph paper. Answers to questions.
ERROR ANALYSIS In this experiment students are not "given" the equation for the pendulum. The purpose is to try to discover the dependence of the period on L and g. So the error equation cannot be written in advance, but only when those relationships are discovered. The uncertainties will show up as repeated measurements are made, and when one constructs the graphs. The size of the uncertainties will be evident from the amount of scatter of points around the fitted curves). This will be considered below, after the graphs and data are analyzed. 6. ANALYSIS Specific points students were asked about. (1) and (2) Look for a clear statement that the period depended on the mass, specifically how, and to what % uncertainty. (3) Value of K = 2π/√g = 2.006 s^{2}/m. (4) Value of n, which should be very near 1/2. (5) Dependence of the period on other variables? Dependence on θ should be noted. (6) Clear statement that no mass dependence was observed. (7) Clear statement that there was no dependence on the size of the bob. (8) Two things must be addressed: The confirmation of the power 1/2, and the specific value of K being in agreement with K = 2π/√g. (9) Determination of g. Should lie within 5% of accepted value. (10) If no angle dependence was observed, the question about angle dependence obviously cannot be discussed. It's unlikely that the data whould show more than the second term of the series. However, if data was carefully taken, the period at an angle of 80° should be (1/4)sin(40°) which is 16% greater than the simple textbook formula (Eq. 6) predicts. This is easily observable. Students have observed it, consistently. (11) How small must the amplitude be for the two equations to be in agreement to 1%? Look at the leading terms in parentheses in equation 7. The terms converge rapidly in size. The second term will be less than 1% of 1 when the angle is less than about 23°. This is probably smaller than the experimental uncertainties. The agreement is good to 5% for angles less than 53°. The agreement is good to 10% for angles up to 78°. So the angle dependence of the period should easily show up in this experiment for angles greater than 50° 7. QUESTIONS (1) Use your graph to calculate the required length of suspension which would give the pendulum a period of 12 seconds.
(2) Why do you suppose we suggested that you use such a strange set of length values? (20, 40, 80, 150, 300 cm, etc.) What moral does this have for experimental analysis and strategy? If you had time to take twice as many measurements, what specific additional values would you choose?
(3) Galileo Galilei (1564-1642) first noticed that the period of a pendulum was independent of amplitude, for small amplitudes. He observed a swinging chandelier in the Cathedral of Pisa [during a dull service, perhaps?] and timed the swings with his pulse. Later, in his Dialogue Concerning Two New Sciences (1638), he describes his understanding of the pendulum:
Check Galileo's calculations against the results of your experiment. Has Galileo got his pendulum theory correct?
(4*) Continuing Galileo's discussion of the pendulum: In the same reference, Galileo writes "Nor will you miss it by as much as a hand's breadth, especially if you observe a large number of vibrations." Here Galileo is giving an estimate of the experimental error. Would you say his error estimate is reasonable? Explain, and state the conditions under which it might be reasonable, or unreasonable.
(5*) You probably noticed that the amplitude decreased with time. The rate of decrease was greater when the initial amplitude was greater. Air drag on the bob is the likely cause of most of this. From your laboratory experience, observations, and results, make some tentative, reasonable hypotheses about the effects you would expect air drag to have on the pendulum's rate of amplitude decrease. To start you off on an appropriate manner of analysis: consider which of the following one might expect air drag to depend on: bob mass, bob radius, bob speed, length of the swing arc. Of those you choose, what mathematical form would the dependence take? For example, if the dependence were on bob radius, would it be KR, K/R, K(R^{1/2}), KR^{2}, or what? State your reasons for each hypothesis. See whether the hypothesis does predict what you observed, if the predicted size of the effect is larger than the experimental error.
(6*) Very often the pendulum departs from motion in a single plane and begins to move in an oval path. Suppose this happened, in a pendulum swinging in an arc of 50 cm, and developed a sidewise component of motion of amplitude 10 cm. Would this affect your results? Woult this alter the measured period? If so, how much? If not, why?
Reaction time By the simple experiment of dropping a ruler between the fingers and catching it, one can find the reaction time to be between 0.1 and 0.2 second. Take the conservative 0.2 second. Then to get 1% error in a time, you'd need to time an interval of 0.2 × 100 = 20 seconds. (NEW MATERIAL) Supplement to Wilson's lab manual. Wilson's is one of the few commercial manuals which takes the same approach to this experiment as we have here. But he doesn't ask for a plot on logarithmic graph paper, so these steps are added. PLOTTING OF GRAPH—METHOD C. Steps 11, 12 and 13. 11. By the use of logarithmic graph paper we can arrive at the value of K and n by a more direct manner than by Method A or B. Read step 8 of Method B in order to understand the procedure and its mathematical justification. 12. Logarithmic graph paper saves us the work of calculating logarithms of our data. First you must determine the range of the data, and how many factors of 10 (cycles) it spans. If, for example, you used pendulum lengths, L, from 25 cm to 150 cm, you will need to plot this on a two cycle log scale. The graph paper has markings for 1, 10, 100 on the scale. Relabel them to read 10 cm, 100 cm and 1000 cm. You might wish to label the subdivisions within each cycle. The first major subdivision above 10 cm would be marked 20 cm, the next 30 cm, etc. The data on period ranges from about 1 second to over 2 seconds. This spans much less than one cycle. Its major divisions will be relabeled 1 and 10. So you will choose log paper which has two cycles on one axis (on which you will plot L) and one cycle on the other axis (on which you will plot T). 13. Plot your data on the logarithmic graph paper. Can you say that the data can be represented by a straight line? If so, draw, with a ruler, the best straight line which fits the data. Select two well separated points on the line (not necessarily data points), and designate them as P_{1} and P_{2} respectively. Then substitute their values into equation (4) and solve for n. (These are the only logs you'll have to take in this method.) Substitute this value of n into equation (1) along with any corresponding values for T and L and solve for k. Now re-write equation (1) in your data record with the proper values of k and n supplied so that T and L are the only variables. Show the computation as a part of your report. 13'. If it happens that you are using log paper in which each cycle on both axes has the same length (measured with a ruler), the slope calculation is even easier. Since lengths on the paper are proportional to the logarithms of the axis markings, the slope of a straight line is just the ratio of the lengths of the legs of a right triangle formed on the hypotenuse defined by two chosen points on that line. Actual data and ResultsOne way to deal with this data is by use of a graph, or a spreadsheet. The following results indicate what may be expected from a linear regression calculation on real data. The data was obtained following a commercial lab manual that allowed students to assume that the period is T = 2π√(L/g) and the objective was to determine the value of g, the acceleration due to gravity. Results from pendulum experiment May 31, 1990, data from students at Lock Haven University. Number Time L T L^1/2 Pendulum data Swings cm sec Summer 1990 x y x^2 x*y Variables 133 120 19 0.902 4.3589 0.81407 3.93284 118 120 25 1.017 5 1.03419 5.08475 80 100 31.7 1.25 5.6303 1.5625 7.03784 89 120 45 1.348 6.7082 1.81795 9.04477 62 100 60 1.613 7.746 2.60146 12.4935 77 120 60 1.558 7.746 2.42874 12.0716 63 120 110 1.905 10.488 3.62812 19.9773 43 100 126 2.326 11.225 5.40833 26.1046 49 120 150 2.449 12.247 5.9975 29.9938 36 100 199 2.778 14.107 7.71605 39.1854 31 100 247 3.226 15.716 10.4058 50.6975 18 60 264 3.333 16.248 11.1111 54.1603 19 60 268.5 3.158 16.386 9.9723 51.7452 Column sums: 13 26.86 133.61 64.4981 321.529 Labels: N Sumx Sumy Sum(x^2) Sum(xy) These values are the terms 4179.88 N*Sum(x*y) in the linear regression 3589.08 Sumx*Sumy formula for the slope 590.801 Numerator of a straight line (see below). 838.476 N*Sum(x^2) 721.621 (Sumx)^2 116.855 Denominator 5.05584 Slope, m Results. Not bad! 1009.13 = g (experimental) 981 = g (Accepted value) 28.1296 Discreprancy 2.86744 % discrepancy The least squares curve fit for the equation Y = mx + b are ERRORS REVISITED Another group of students, using similar data, used a graphical approach. They graphed √L vs. T. L is surely the most precise of the two, so treat the uncertainties as being entirely in the measurement of T. The deviations of T from the fitted line averaged about ± 0.1 second. The slope, S = (ΔL)/T was 5.2 cm^{1/2}/sec. S = (√g)/2π. So g = (2πS)^{2} = 1067 cm/s^{2}, which, compared with 981 cm/s^{2}, has a discrepancy of 8%.
From the scatter of points on the graph, they judged the slope to have an uncertainty of about 5%. That uncertainty is doubled when the slope is squared (giving 10%), so their value of g was reported as 1067 ± 106 cm/s^{2}. The discrepancy is less than the estimated uncertainty. So this result is within expectations. © 1998, 2004 by Donald E. Simanek. |