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

About your lab report:  You'll submit both a Word document and a Graphical Analysis 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. Read section 13-6 before starting this lab.
  2. Complete M08: Period of a System in Simple Harmonic Motion. The method of analysis is similar to that of L09.
  3. Assemble the following materials:  stopwatch, a couple meters of string, a compact weight to hang from the string as the pendulum bob, a support for the pendulum, tape, meter stick.
  4. Read about Design Considerations below before setting up your own experiment.

Design Considerations

We generally use 9.80 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, plot a graph of T vs. L, fit the graph to a square root function, and determine the value of g from the coefficient of fit.  The method is similar to the one you carried out in M08 to find the value of the constant in the formula for the period of a spring.

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 less than an inch 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 constant.  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 constant.

  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.  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

You should be ready now to set up your equipment to take data.  In addition to setting up the equipment, prepare a data table that will allow you to present all of your data in an organized way.  Think about how you will do this in order to provide a logical display and make the table easy to read.  Before beginning to record data, practice your timing technique.  Come up with a technique that you think will minimize your starting and stopping errors.  When your preparations are complete, take your data.  If you're working with a partner, you can both use the same pendulum, but you must do your own timings.  Also, check each other out on measuring the length of the pendulum. 

Analyzing the Data

Use Graphical Analysis for analyzing the data.  The process should be similar to what you did for M08.  Reread the goal of L09 to make sure that your analysis is directed toward that goal.

Writing Your Report

While your analysis will be submitted in a Graphical Analysis file, your report will be a 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:  Don't derive the theoretical formula for the period of a pendulum, but do show by dimensional analysis that the formula must have the form T = C(L/g)½, where C is a constant. In order to review the method of analysis, see M08 and section 1.3 of the text. Also describe the method of graphical analysis that you will 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.

Graph/fit, matching table and equation of fit:  While your graphical analysis will be in a separate file, 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 percentage uncertainties and deviations. In addition, since you have an accepted value for g, calculate the percentage difference between that value and your measured value. See the FAQ for labs to review these methods.

Qualitative:  Write a qualitative analysis of error. 

Submitting your report

Name your GA and Word files in the standard way and submit them as requested.