THE SCIENTIFIC METHOD
EXPERIMENTS WITH THE SIMPLE PENDULUM
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RATIONALE:
In the physics laboratory we hope to acquaint you with some of the techniques of systematic investigation in physics, to give you some familiarity with the instruments of investigation, and to provide an opportunity for you to make such investigations on a limited scale. This systematic procedure of approaching problems is often termed the scientific method. Hopefully during this term you will cultivate enough familiarity with the scientific method so that the method will become yet another tool in your ability to develop answers and responses for your future endeavors. History shows the scientific method is an extremely successful procedure for solving a wide variety of problems. The physics laboratory is a place to learn physics. Its function is not just to illustrate principles that have previously been established as "true" in the lecture sessions. With this in mind, you should not be surprised to find the laboratory dealing with topics not as yet discussed in class. The present experiment deals with factors affecting the period of a simple pendulum--a topic that will not be discussed in class for several weeks, but which serves well to illustrate laboratory procedures.
SPECIFIC OBJECTIVES:
When you have completed this laboratory exercise you should be able to: (1) distinguish between independent and dependent variables in an experimental setting; (2) use the balance and meter stick to measure mass and length; (3) identify the least measure of an instrument and calculate the expected error associated with the use of the instrument; (4) define amplitude and period as associated with periodic motion and give the units of measure of each; (5) construct a suitable data sheet for recording experimental data; and (6) given corresponding values for the variables X and Y, plot an acceptable graph of Y = f(X). Most importantly, you should understand how to conduct an experiment to expose the relationship between dependent variable Y and the independent variable X when other independent variables W and Z are also present. It is in the understanding of this concept that the scientific method has its greatest strength and usefulness.
EXPERIMENTAL ACTIVITY:
A simple pendulum is one in which the mass is concentrated in a volume which is small compared to the other dimensions of the problem. A small spherical mass at the end of a string makes a good approximation to such a pendulum. The motion of a pendulum is repetitive. In one complete swing from left to right and back again, the pendulum is said to go through one cycle. All subsequent motions are a repeat of this cycle. The PERIOD of the pendulum is the time required for it to go through one cycle. It is the time required for the pendulum to move from any point on its path back to the same point with motion in the same direction. The AMPLITUDE of the pendulum measures how far it swings away from the vertical. The following figure illustrates the proper measurement of amplitude. This is conveniently expressed as an angle in degrees. For the purpose of this experiment the MASS of the pendulum is defined operationally as what you measure when you "weigh" the pendulum on a balance.
Among the factors that might affect the period of a simple pendulum, we can list the mass of the pendulum, the length of the pendulum and the amplitude of its swing. As these are easy to control experimentally, we shall confine our attention to these.
If we are to investigate the effect of any one of these variables on the period of the pendulum, the remaining variables must be controlled; i.e., they must not be allowed to change during that portion of the experiment. Suppose we start with an investigation of the effect of length upon the period. This means that we choose a pendulum of fixed mass, allow it to swing through angles of the same amplitude, and observe changes in the period due to changes in the length of the pendulum.
Prepare a pendulum whose length is between 20 and 25 cm. Set it swinging with an amplitude of about 20 degrees, and determine its period. To improve the accuracy of timing, time 10 complete cycles, and then the time of one cycle is just that total time divided by the number of cycles timed. You should also check any measurement for consistency (repeatability) by repeating the measurement 3 or 4 times. Any large variation in the measurements is an indication of a problem that needs to be resolved before you go on in the experiment.
You will need some systematic way of recording your data. The best way is to make a DATA TABLE that gives a place to record your observations in an organized manner. A suggested data table is shown below. Learning good ways to organize your data will help you to work faster and make fewer mistakes in the laboratory. In the future you will be responsible for determining the shape and style of your own data tables.
PERIOD OF PENDULUM OF LENGTH L
|
MASS = 89.45 g |
AMPLITUDE = 20° |
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|
LENGTH (cm) |
TIME FOR 10 CYCLES (sec) |
PERIOD (T) (sec) |
|
|
27.32
38.16 |
10.5 10.4 10.5
12.4 12.2 12.5 |
1.047
1.237 |
|
Working carefully now (but not too slowly) you need to increase the length of the pendulum to a new value and determine the period in the same manner. Record the values on the data sheet. Repeat the measurements until you have taken observations for five different lengths, the last of which should be longer than 100 cm. Remember that the mass and the amplitude must remain the same throughout this series of observations.
Proceed now to study the effect that different masses have on the period of the pendulum. Recall that you will have to keep both length and amplitude constant during this set of observations. Use at least 5 masses, and record your data accurately in a new data table.
Finally, conduct an experiment to determine the effect of amplitude on the period. Use at least 4 different amplitudes, with the largest being more than 60 degrees. Record your data in a third data table similar to the two previous ones.
ANALYSIS OF RESULTS:
The significance of experimental data is often more apparent when it is presented in visual (or graphical) form. Prepare a graph of the period (vertical scale) versus the length of the pendulum from the information in your first data table. Refer to the instructions in the front of this manual for aid in preparing the graph, if necessary. In the discussion portion of your report, answer this question: What can you tell from this graph about the relationship between the length of a pendulum and its period? Try to avoid letting prior knowledge of the pendulum bias your answer. Can you tell from this graph the nature of the mathematical function that describes the relation between L and T?
To see the relationship more clearly, make a new, fourth column on data table 1 in which you enter the square of the period. Now make a new graph of period squared versus length. What is the form of this graph? What can you say about the relationship between T2 and L? Can you now determine the relation between T and L?
Examine the data from the second and third data tables. Determine the relationship between the variables for both sets of data. If necessary, make graphs to help your analysis. When you know both of these relations, then you are ready to summarize the results of all your observations.
DISCUSSION:
It is shown in you textbook that the period of a simple pendulum (vibrating with small amplitude) is given by:
where g is the acceleration of gravity and is about 9.81 m/s2. Are your results consistent with the prediction of this formula? Explain.
Assuming that the formula above is valid, we can manipulate it into the form:
and 
Now using one of the data points from data table 1, calculate an experimental value for g. Show your calculation. Then repeat the calculation for the other data points in your table. You may wish to make an extra column on the table to record your values. Find the average of the five values of g, and determine the percent error between your average value and the reference value of 9.81 m/s2. Show your calculation.
FINAL SUMMARY:
In your report summary, include the average value g. What is the relationship between the period and the length? between the period and the amplitude? between the period and the mass? What other conclusions can you draw about the pendulum?
QUESTIONS: