L09.  Measuring the Acceleration Due to Gravity with a Simple Pendulum

About your lab report:  You'll submit both a word-processed report and a Logger Pro file.

Goal

Measure the acceleration due to gravity to 3 significant figures by using the relationship between the length and period of a simple pendulum

Preparation

1. Assemble the following materials:  stopwatch, a couple meters of string, a steel ball to serve as the pendulum bob (look in your lab kit for this), a support for the pendulum, tape, meter stick.

Design Considerations

We generally use 9.80 or 9.81 N/kg (or m/s2) as the value of the gravitational field, termed g.  This is also the value of the acceleration of a free-falling object at the surface of the Earth. The actual value, however, depends on your location. The purpose of this lab is determine--subject to certain constraints--the value of the gravitational field at your school.  The primary constraints are that you must use a simple pendulum, your timer must be a manual stopwatch, and you're expected to achieve 3 significant figures in your value of g.  However, you won't be penalized for achieving less than three figures assuming that you've designed and executed your experiment well with the materials that you have available.

You've measured g in a previous lab using a different method.  In L04, you used video analysis to measure the acceleration of a lead-weighted racquetball in a long fall.  Air friction was small enough in that situation that you were able to obtain a value for g close to what was expected.

The present lab provides a standard way to measure g to high accuracy and, in the process, to see the interplay between theory and experiment.  The period, T, of a simple pendulum in the small-angle approximation is T = 2p(L/g)1/2, where L is the length of the pendulum.  One can use this formula to calculate g, given measurements of T and L.  The general plan will be to measure T as a function of L, re-express variables appropriately to obtain a linear fit, and determine the value of g from a coefficient of the fit.  You've used this method many times before.

The precision and accuracy of the measurement of g with a simple pendulum depends on the smallness of the angle as well as the measurements of T and L.  Here are some design considerations:

1. The initial angular displacement of the string from the vertical should generally be less than 10°.  The motion of the pendulum closely approximates simple harmonic motion for small angles, and the formula T = 2p(L/g)1/2 applies.

2. The string must have a sturdy support.  If the support wobbles, that can affect the period.

3. The weight (bob) on the end of the string must be compact and dense.  If the bob is dense, the effects of air drag will be kept small.  (A small amplitude swing has a similar effect, since the velocity of the bob remains low.)  By compact, we mean that the dimensions of the bob must be small in comparison to the length of the string.  The bob needs to act as nearly as possible like a point mass.  A metal ball a few centimeters in diameter works well.

4. Since a real bob isn't a point, the length of the pendulum is measured from the point of support to the center of mass of the bob.  You may need to estimate the position of the latter if the bob doesn't have a symmetrical shape like a sphere or cylinder.

5. In order to determine g to 3 significant figures, you need to measure T and L to at least 3 significant figures.  If you measure L to the nearest millimeter, that will give you 3 significant figures for distances under a meter.  Achieving sufficient precision in the measurement of T takes some consideration of how you will time.  The length of a seconds pendulum, that is, one with a period of 1 second, is a quarter of a meter.  If you measure the time for 1 cycle of such a pendulum with a manual stopwatch, the tenths digit will be uncertain even though the stopwatch may give a readout in hundredths of a second.  The uncertainty arises from starting and stopping errors.  A simple way to reduce the effect of such errors is to time several consecutive cycles.  Suppose, for example, that the starting and stopping error is 0.2 s.  if you time 10 cycles from start to finish, that 0.2 s error is one-tenth as much per cycle, or 0.02 s.  This has the effect of adding one significant figure on to the measurement of period.  It's important to realize that this method only works well when the thing being measured is uniform.  That's another reason to keep the initial angular displacement small.  As the amplitude of the swing decays over 10 cycles, the period will remain very nearly the same.

6. A rule of thumb from previous labs is that one generally gets better accuracy with larger measurements.  With the pendulum, the longer the string, the longer the period will be.  Thus, set up your pendulum so that its greatest length is as much as you can accommodate with your lab arrangement.  If you can get 2 meters, that's good.

7. In order to show that you've achieved 3 significant figures in your measurement of period, you need to show that you can reproduce the measurement to that precision.  That requires taking several time trials for each length and calculating percentage deviations.  Five trials is generally enough to be convincing.

8. In order to obtain a good fit for T vs. L, you'll need several data points from small lengths to large.  At the lower end, you're limited by the fact that the dimensions of the bob need to be small in comparison to L.  For a bob of 2 cm diameter, making L less than 10 cm isn't advisable.  At the higher end, you're limited by how much vertical space you have and where you can hang the string from.  Go for at least 5 different values of L.  Space them closer for smaller distances, because the T α L½ relationship that you expect increases faster for smaller L.

As you can see from the above, there's much to think about in designing an experiment to give precise and accurate results even with the simplest equipment.  While we spent much time discussing experimental design above, the remaining instructions are brief.  By this time, we expect you to be well-versed in collecting and analyzing data.

Setting Up and Taking Data

Analyzing the Data

Use Logger Pro to analyze the data.  Use the standard process of re-expressing variables to obtain a linear fit. Provide a matching table and equation of fit, and show how you determine the value of g using the results of the fit. The correct method is to use a numerical coefficient from the fit and equate that to the expected physics expression for that coefficient. You obtain the latter from the theoretical expression for period as a function of length. If you don't use this method, you'll receive no credit for the analysis. By this time in the course, the method should be second nature to you.

While you'll submit your analysis in a Logger Pro file, your report will be a word-processed Word document.  This is to be a complete report as described in the lab guide.  All sections listed in the guide are required. Here are some specifics about particular sections.

Theory:  In this section, describe the method of graphical analysis that you use to determine the value of g.

Method: Describe specifically how you addressed the various design considerations described above in setting up and carrying out the experiment. Make a list lettered as above (a-h). Each response must be complete in of itself without having to refer to these instructions in order to see what design point you're addressing.

Data:  Present all your data in a logical and easy-to-read format with complete labels. You will need to include deviations and means.

Graph/fit, matching table and equation of fit:  Copy and paste your graph, including the fit, in your Word report. To review how to write a matching table and equation of fit, see the FAQ for labs.

Sample Calculations:  Show how you use one of the coefficients of your fit to determine a value for g.

Error Analysis

Quantitative:  Carry out a quantitative error analysis for your determination of the acceleration due to gravity.  Doing such an analysis has been a part of several labs up to this point, and we expect you to use techniques learned previously.  You must calculate deviations and mean percentage deviation. In addition, since you have an accepted value for g, calculate the experimental error between that value and your measured value. See the FAQ for labs to review these methods.

Qualitative:  Write a qualitative analysis of error. Review how to do this in the lab guide.  The non-descriptive phrase human error has been showing up more and more in recent lab reports. This always results in a score of 0 for the associated question.