Experiment: The Pendulum


Experiment Pendulum

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Pendulums have been used for a variety of purposes including time keeping, and as accelerometers, seismometers, and gravimeters. Galileo is generally credited as being the first to study properties of the pendulum scientifically. The most interesting property is called the period of a pendulum–the time for one complete left an right oscillation. Galileo found that the period of a pendulum is independent of its mass, and that the period is independent of the angle through which it oscillates, at least for small angles.

In this experiment you will investigate the relationship between the length of a pendulum and its period of oscillation. This will be accomplished by using an Arduino sketch (program) that automatically times the pendulum’s swing as it interrupts light that is striking a light sensor. Data from the sketch can then be imported into an Excel file (provided) for student analysis.

Duration: About one class period

GRADE LEVEL
High School (ages 14-17)
College/University (age 18+)

DIFFICULTY
Intermediate

SUBJECT
Science

MODULES & ACCESSORIES USED (6)
light sensor (1)
power (1)
bright led (1)
Arduino (1)
wire (2)

OTHER MATERIALS USED (10)
Mounting Board 2
Piece of wood about 8″ x 1.5″ x 0.75″ 1
velcro 1
Opaque soda straws 4
Sewing needle (approx. 2 and 1/2 inches long) 1
Ring Stand 1
Burette clamp 1
Computer with the Arduino IDE installed 1
Metric ruler 1
Excel spreadsheet software 1

LESSON GUIDE

STEP 1 : Prepare the Pendulums

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The pendulums for this experiment will be opaque soda straws that have been cut to different lengths. The straws need to be opaque so that they will block the light each time they swing by. The suggested lengths are approximately as follows: 6 cm, 11 cm, 16 cm, 21 cm, and 26 cm. The pivot point for each of the pendulums will be a 2 and 1/2 inch sewing needle that has been pushed through near the top. See the picture below. Students should be cautioned to use care when handling the sharp sewing needles.

STEP 2 : Prepare the Circuit

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The circuit consists of the following modules in the order given:

power>bright led>wire>light sensor>wire>Arduino

See the photo below.

The power and bright led are on one mounting board. The sensor is on another mounting board. Velcro is used to stick the two mounting boards to a small wood block to keep the boards stable and upright. The light sensor must be set to light (not dark) and set to maximum sensitivity (fully clockwise). You will probably find it best to do the experiment in a semi-darkened room with lights off.

You should cover the red led on the power module with black electrician’s tape so that it will not interfere with brightness readings during the experiment. The wire module must be connected to the Arduino a0 input. Also, the Arduino must be connected to your computer via the micro-USB cable provided with the Arduino.

STEP 3 : Practice Swinging the Pendulums

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The picture below shows the apparatus set-up. A ring stand with a burette clamp is adjusted so that the pendulum interrupts the light beam from the bring LED to the light sensor as the pendulum swings back-and-forth. The sewing needle at the top of the pendulum acts as a pivot point for the pendulum. To get used to working with the apparatus, you should practice swinging the pendulum a few times before actually collecting data. There is a link to an m4v movie to the right which shows the pendulum swinging.

It is suggested that when performing the experiment, you begin with the longest pendulum and then proceed to the shorter pendulums.

STEP 4 : Ready the Arduino Module for Data Collection

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Note that the Arduino module’s micro-USB cable must be connected to your computer, and the computer must have the Arduino IDE software installed. THE ARDUINO SWITHCES SHOULD BOTH BE SET TO ANALOG.

Power up the Arduino module. Start the Arduino IDE software. Select Tools>Board>Arduino Leonardo. Then select Tools>Serial Port and select the serial port that the Arduino will use for communication. This will depend upon whether you are using a Windows, Mac, or Linux machine. Open the sketch file called Pendulum.ino. Upload the sketch to the Arduino module. You will see the yellow rx/tx LEDs blink on the Arduino module while the sketch is uploaded. After the blinking stops, you will then have 15 seconds to start the serial monitor by selecting Tools>Serial Monitor from the IDE and pull the pendulum aside so that it is not blocking the light from the LED. After the 15 seconds have elapsed, you can then release the pendulum. Data collection will automatically begin and proceed for 10 seconds, with results appearing in the Serial Monitor.

After the 10 second data collection period is complete, the Serial Monitor will look similar to that in the image below.

STEP 5 : Transfer the Data to the Excel File

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When you are done collecting data timing for a pendulum, then you can transfer the data from the serial monitor to Excel. Here is how to do that. First, uncheck Autoscroll in the bottom left corner of the serial monitor. Use your mouse to drag and select all of the data beginning with the data line containing the column headers “Time Brightness”. Open the attached Excel .xlsx file called Pendulum_Template.xlsx. Note in the bottom left corner of the Excel window that there are five worksheets, one for each of the pendulum lengths. Make sure that you have the worksheet selected for the length of the pendulum whose timings you just obtained. Right-click on cell A1 in this worksheet and select Paste.

You should see the data in the Excel workbook on the far left. In addition you will see a chart displaying a graph of brightness versus time. Your chart should look similar to the chart displayed below. Note that the time axis is in seconds.

The figure below shows four complete periods (left-and-right oscillations) of the pendulum. Remember that the pendulum blocks the light twice for each complete period, once while traveling to the right, and once while traveling back to the left.

You can determine the time when the light is blocked the most by placing your cursor over the bottom point. A tooltip will pop up in Excel telling you the time and brightness. For higher accuracy, it is suggested that you get the time for ten complete periods, and then divide by 10 to get the time for the period of the pendulum’s oscillation.

You will notice above the chart on the Excel worksheet are two cells highlighted in green. You need to record the length of the pendulum, and the period in those green cells. Length must be in meters, so for example, if you have just done a pendulum whose length is 25.3 cm, you would record the value as 0.253. Length should be measured from the pivot point to the bottom of the pendulum. Period should be in seconds. As an example using the chart shown below, the period would be (6 – 3)/4 = 0.75.

You should repeat these procedures for the remaining four pendulum lengths. When you are ready to collect data for a pendulum, simply re-upload the sketch to the Arduino module. Again, you will have 15 seconds to open the serial monitor and move the pendulum so that it is not blocking the light from the LED. Release the pendulum when you see the “Time Brightness” header at the top of the serial monitor window. Transfer the data from the serial monitor to the appropriate worksheet (depending on the pendulum’s length) in the Excel file. Calculate the period, and then record the length and period in the two highlighted green cells above the chart.

STEP 6 : Data Analysis

Now for the most important part of this experiment–data analysis!

Select the worksheet entitled “Analysis” from the bottom left corner of the Excel window. You will notice that Excel has automatically copied your values for pendulum length and period into a table in the upper left corner of the worksheet. Excel has also prepared a chart of pendulum period vs. length, based upon your data.

The graph does not appear to be a straight line, but rather, curved, with decreasing slope as length increases. This suggests that there may be a power relationship between period and length. As a result Excel has been asked to plot a best-fit power relationship to your data. The resulting equation is shown in the upper left corner of the chart. y is the dependent variable (period) that is graphed along the y-axis. x is the independent variable (length) that is graphed along the xx-axis. Note the power for x. You should find that it is close to 0.5. This means that the period of the pendulum is proportional to the square root of the length, since square root is equivalent to the power 0.5.

STEP 7 : Determine the Acceleration Due to Gravity (Optional)

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Historically, pendulums have been used to determine the acceleration due to gravity. With your data you, too, can determine the acceleration due to gravity.

Note the constant of proportionality in the equation for the power curve fit of period to length shown in the chart in the “Analysis” worksheet. Physics tells us that for a pendulum that is a rod (as in the case of our straws) and not a weight hanging from a string, the formula below relates the period T and length L of the pendulum.

Using a little bit of algebra, get this equation in the form T=constant times square root of L. Then equate the constant to the constant of proportionality from the equation in your experiment. You can then find the value of g, the acceleration due to gravity.

How does your value of g compare to the accepted value of 9.8 m/s/s? What is the percent error? What do you think are the contributing factors for this error?