Law of Lever and Pendulum Experiment

The principles of the pendulum have been known for centuries. One application has been in time-measuring devices such as the “grandfather clock”. Another is the “swings” we all played on as children. In our experiment we will vary the length of the pendulum to determine its effects on the period T (time for one full cycle of oscillation). We will also find that the magnitude of the gravitational constant “g” can be determined by experimental means through use of a pendulum.

Our pendulum will consist of a light string with a small weight attached. A small steel “nut” of about ½ inch in diameter works well. It can be a little larger or smaller. Some other small mass may be used if a string can be easily attached to it. A thin string works better than a thick cord. A firm support must be used that won’t move during oscillation of the string.

I usually use a paperclip (about 1.75 inches long) as a fulcrum point. First, bend the end of it slightly out and then up toward you at about 30 degrees (see drawing on data sheet) so that it slightly resembles a fishhook. Use “Scotch tape” to attach the flat portion of the paperclip firmly to the overhead of a door facing. Attach it so the bent end hangs slightly below the door facing. This enables the pendulum string to swing freely back and forth through the doorway when a small loop is looped over the hook. This arrangement also facilitates changing the length of the pendulum. After working with one length, a new loop can be tied in the string and hung over the hook to begin again.

We will do the experiment at least five times. Lengths of about 0.3, 0.4, 0.6, 0.8 and 0.9 meters should be used. Don’t try to make precise measurements of these lengths before you hang the string. Make a modest try and then be happy with what you get. Next, hang the string with its small mass attached. Then precisely measure the length from the bottom of the paperclip (or wherever the string is attached) to the center of mass of the weight. That’s why a “nut” works well. The center of mass is at the center of the hole and can be read on the meter stick when looking “through the hole”. Measure the length to the nearest millimeter and log this information on the data sheet.

The time required for the pendulum to go through one complete cycle is called the Period (T) and is measured in seconds (to 1/1000 of a second). Use a stopwatch to time for ten complete oscillations (cycles) and divide by ten to get the number of seconds per cycle (T). Repeat the process for the same length and use the average of the two values for the period of the pendulum. If an appreciable difference exists, repeat a third time to see which value is correct. Use only the two “good” values for calculating the period.

When setting the pendulum into motion (through the doorway), displace it by only about 10 degrees. The most accurate time measurement is obtained by standing in the doorway, facing the oscillation as it goes back and forth in front of you (left to right, etc). Begin and end your time measurement as the mass passes through the “vertical” which can be seen on the “door facing” in the background. Remember, one oscillation is the time from when it passes through a point until it next passes that point in the same direction. Due to reaction time, “lead” your operation of the stopwatch so that you begin and stop timing exactly as it passes through the vertical.

Repeat the procedure for each separate length of the string. Then use your calculator to find the “logs” (to 1/1000th ) of the various numbers and the value of “T squared” (to 1/100th) for each string length. Don’t “round off” beyond this. Enter all of these values on your data sheet and make two graphs of the results as follows.

GRAPHING “LOG T” VERSUS “LOG L”

One of the goals of the experiment is to learn how to prepare a good graph. You could graph it all in one square inch on your paper but it would be a terrible presentation. Experiment with various choices of scale and positioning of the graph on the paper to best display your experimental data. Choose a scale that will use nearly a complete sheet of paper for your graph. Label the axes as “log T” (on vertical axis) and “log L” (on the horizontal axis). The best technique is to make a small firm “dot” for each plotted point on the graph and then make a very small circle around it.

The graphed Log T and Log L data points should fall in a fairly straight line. Then draw the best straight line possible through these points. If you have one “point” that doesn’t line up “well” with the others, it should be discounted as an error was probably made in some phase of the experiment (make sure it was properly graphed!). Next, find the slope of the line as you learned in algebra. See my sample graph sheet for my computation of slope. The slope should be approximately 0.50. Next, read the point where the line crosses the “Log T” axis. Enter that number on your calculator and take the inverse log of it. (On my TI-30X calculator, I enter the number, press “2ND” and then “LOG”). Your answer should be approximately 2.01.

GRAPHING “T- SQUARED” VERSUS “L”

Use similar graphing procedures and find the slope of the graph as before. Equate the resulting slope to the quotient of “4 pi squared” divided by the letter “g”. Then solve this small equation for “g”. You should get approximately 9.8 m/s squared.

One of the attachments to this experiment contains the equation showing the period T to be a function of the length L and the gravitational constant “g”. If you have achieved reasonably good results in your experiment, you have substantiated the validity of this equation. Some small errors will always exist in an experiment. In this case, errors could be made in measuring the length of the pendulum, reaction time errors in use of the stopwatch or using an excessive angle of oscillation. Some friction will always exist, both at the pivot point of the string and the friction due to air resistance. Usually the sum of these errors is small enough that good results are still achieved. This proves once again that some of the secrets of the physics of our universe can be learned through use of the most common items. What are your thoughts after having done this experiment? What did you learn? Did you make neat graphs?

SENDING RESULTS:

1. Worksheet for determination of “g” values”

2. Graph of Log T vs Log L

3. Graph of T squared vs length L

Having this submitted material enables me to

(1) evaluate how well you did the experiment

(2) provide feedback for any existing errors