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INTRODUCTION

   The period of a pendulum is determined by its length. You will compare the measured period of a real pendulum (spherical ball suspended from a string) with the expected period of an ideal pendulum (point mass suspended from a massless string), as determined from Newton's second law. You will also use your data to determine the acceleration due to gravity, g.

EXPERIMENT

   You are given a pendulum with a support with which you can adjust the length of the string, and a photogate. Use the small cylinder attached to the string to interrupt the timer's light beam. Set up the photogate so that the beam is interrupted when the pendulum is at rest. The photogate is set sideways so that the swing angle, θ, is kept small (a small angle is assumed during the derivation of the equation found below).

Pendulum Length (): The analysis assumes the length of the pendulum to be the distance from the top clamp to the center of mass of the ball. It is easiest to measure the length of the string to the top of the ball, then later add the radius of the ball.

Period (T): In the "PEND" (pendulum) mode the timer will display the time interval for one full swing, the period T of the pendulum (the light interruption during the backswing is ignored). Make sure the timer is also set to 1 ms.

1. Using a vernier caliper, measure the diameter of the ball along the axis the string passes through. Calculate the radius of the ball.

2. Create a data table in your report (at the top of a fresh sheet of paper) with the following headings:

   Lbottom (cm)

   Ltop (cm)

   Lstring = | Lbottom – Ltop | (cm)

   Period, T (sec)

   <T> (sec)

   = Lstring + rball (cm)

3. For a string length about 90 centimeters long (the longest you can probably use on the table), measure the pendulums' period (if you want to make the experiment more interesting, let the pendulum hang over the edge of the table; you can get a pendulum length up to 2 meters!). The period will lengthen as the amplitude of displacement increases, so keep the angle less than about 10 degrees. Repeatedly measure and record the period by pressing the reset button, until you are satisfied with the period for this length.

4. Calculate the average period, and the length of the pendulum. Plot this data point on a graph of Average Period as a function of Pendulum Length.

5. Shorten the string length, measure the period and calculate the pendulum length. Continue for intermediate lengths down to the smallest you can measure (four or five centimeters). Plot as you go, using your graph to "tell" you which length to choose next.

From Newton's second law, the expected period is:



This expression can be written as:

.

Note the units of the constant c are when g = 980 cm/s².

   The theory follows a power function: y = axⁿ, where n is 0.5 (because of the square root). If we use Excel for our analysis, it will attempt to adjust n as well as a to get the best possible fit; unfortunately, anything other than n = 0.5 produces lousy results. So, to improve our analysis, we will use a different program, KaleidaGraph, which is better suited to the task of analyzing our data. (Note: we could still use Excel for our analysis if we linearize the data — i.e. produce a straight-line plot — and plot T² vs. ).

6. Use KaleidaGraph to create a table of your data containing and <T> (Note: Don't include 0, 0 in your table!). It's important that you label the data columns (by default, they're labeled A, B, C, etc.). Double-click on the first column title ("A") to open a dialog box; change the title for columns A and B to L and <T> (be sure to include units!!), then click OK.

7. Create a graph of <T> vs. as follows:
   1. From the Gallery menu, choose Linear, then Scatter. In the "Plot" dialog that appears, click the radio buttons next to L and T to graph them on the X and Y axis, respectively (now you see why it's important to label the columns!). Click the New Plot button.
   2. Select the Curve Fit menu; you'll see many types available. Choose General, then Square Root (Note: this is a custom fit added by your instructor). Click the box next to <T>, then click OK. A curve will be fit to your data, and a table of results will appear. Click and drag the table to move it on your graph so that it doesn't obscure the curve (Note: if the table of data does not appear, look under the Plot menu, and select Display Equation).
   3. Before printing your graph, you should fix it up a bit.
      1. Turn off the legend by choosing Plot, Display Legend.
      2. Create a proper graph title (it probably says "Data 1" now); double-click the existing title, edit it, and click OK.
      3. Extend the graph axes so that they include the origin. From the Plot menu, choose Axis Options. In the dialog box that appears, choose the X or Y axis from the pulldown menu, then change the Min value to 0. Click OK when finished. (If the line does not extend beyond the first and last data point, from the Format menu, choose Curve Fit Options, then select Extrapolate Fit to Axis Limits).
      4. With the graph selected, choose the Text tool (it looks like an uppercase T), and add your names to the graph. If you wish, you can format the font, size, and color of the text.
      5. Choose Print Graphics from the File menu to print the graph.

8. Use your measured value of the constant c to calculate an experimental value for the acceleration due to gravity, g.

DISCUSSION

• Report your measured values of c and g.
• Compare the expected and measured values of g.
• Discuss how well your data fit the theory. Considering that the theory is derived based upon an ideal pendulum, do you think it sufficiently describes the behavior of a real pendulum?

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©	St. Lawrence University	Department of Physics

Revised: 25 Jun 2003

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