Introduction
In this experiment you will investigate how the period of a simple pendulum, i.e. the time it takes to undergo one complete cycle of its motion or oscillation, depends upon its length.

Apparatus
stopwatch
stand and clamp
pendulum bob and string
meter stick or measuring tape

Procedure
Using the stopwatch, measure as precisely as you can the time it takes for 10 complete cycles of the pendulum's motion at some fixed length. To keep the motion "simple," start the pendulum swinging from an initial angle of no more than 20 degrees. You needn't begin timing on the first swing, you can set it in motion and then begin timing when it seems comfortable. This may increase the precision of your results by decreasing the uncertainty associated with the times at which the pendulum is released and the stopwatch is started. After determining the time for 10 oscillations, let a partner have the stopwatch, and measure again. Finally, let the other partner take the stopwatch and repeat the measurement. This is called the "alternating partners approach." Now use a meter stick or tape to measure the length of the pendulum, from the pivot point to the center of the bob (the location of the "center of mass").

Question 1. Did all the partners' results agree? Comment.

Enter your data directly into an Excel spreadsheet. The spreadsheet is a matrix of rows and columns that enables you to quickly plot data and perform calculations. You can enter data into a particular cell by left-clicking on that cell with the mouse, and then typing in values using the keyboard. Let columns A, B, and C represent the three sets of times, one for each partner. Let column D contain the length data. Use the first row to label the columns.

Repeat the above sequence of three sets of time measurements for each of at least 7 different pendulum lengths (hence, 7 rows of data in the spreadsheet). Be sure to include a wide range of lengths, say between approximately 5 cm and 2 meters, including these extremes. Remember to start the pendulum swinging from a relatively small angle, less than 20 degrees. Continue to use the alternating partners approach, or split up duties however you like.

Analysis
1. Calculate the average value of the period for each pendulum length by averaging your three results for each data set and dividing by 10. You can easily put these results in Column E by doing the following manipulations:

a) Click on cell E2 and type "=average(a2:c2)/10" without the quotes, followed by the "enter" key. The "=" sign defines the cell's contents in terms of what follows it, the term in parenthesis defines all the time data in row 2, and "/10" divides by 10.

b) To repeat this calculation for the rest of the data, left-click again on cell E2 to highlight it, then move the pointer to the lower right hand corner of the cell. It should turn into a solid black cross that you can then click on and "drag" down the E column to the last row that contains data. Then let up on the mouse button, and the column should fill with the appropriate calculations.

2. Plot your data, with the period on the vertical axis, and the length on the horizontal axis. To do this, first click on cell D2, which contains the first length you measured, and while holding the left mouse button down, drag the cursor over all the cells in columns D and E that contain data. They will thus be highlighted in black. Then click on the "chart wizard" icon (it looks like a bar graph with bars of different colors), and then click on "XY (Scatter)" plot type. Then make a "line" plot, i.e. select the icon that shows thin lines with no data points, and then click on "next." This shows you what the plot will look like. Click on "next" again and you can enter a title and label the axes by clicking on the appropriate boxes. Then click "finish" to display the plot on your worksheet. You can then move it to an appropriate place where it doesn't obscure your columns of data and calculations.

3. Make a second graph, this time with the square of the period (i.e. period x period) on the vertical axis and length again on the horizontal axis. It's easiest to copy column D (lengths) onto column F by clicking on cell F2, typing "=D2" and then dragging down the F column as before. Then you can click on cell G2 and type "=E2^2" which will calculate the square of the value in cell E2, that is, the period squared, and put it in cell G2, then drag to fill that column. Then highlight columns F and G as before, and make a similar plot, labeling the axes, etc. Note that when making a scatter plot in Excel, the highlighted column on the left is plotted on the "x" axis, and the one on the right on the "y" axis. Again, move (and enlarge) this plot so that your data and the two plots fit nicely on a single page (you can use the "view" function to check this out).

Finally, make a copy of the worksheet with data and plots, for each partner. You can print all the copies, or print one and photocopy the others later. Each partner needs to attach a copy to his or her lab notebook. Also, save your worksheet in the appropriate lab section folder on drive C:.

Question 2. Qualitatively, how does the period of a pendulum appear to depend upon its length?

Question 3. Describe the shapes of the plots made in parts 2 and 3 above. What type of quantitative conclusion about the motion of a pendulum can you make from your observations? How do you think these might fit into a theoretical description of this motion?

Question 4. Use your data to determine the length of pendulum required to generate a period of 1 second. Can you think of a technological application?

Question 5. Reflect upon the "scientific method" you employed in lab today and in your analysis, and identify the different stages of your activities in this process. We'll discuss the notion of a "method" in class on Thursday when we turn to the Barbour reading.

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