Grandfather's Clock
Objective: To investigate how the period of a pendulum varies with its length.
Procedure:
1. Hook your pendulum bob to the string and slide the other end of the string into the pendulum clamp.
2. Slide the string through a pendulum clamp so that the length of the string to the center of the bob is 75 cm.
3. Adjust the height of your photogate and/or pendulum clamp such that when the pendulum is in its equilibrium position, it blocks the photogate's sensor. You may need to raise or lower the entire horizontal rod in order to do this (note that this does not change the length of the pendulum).
4. Make sure that your photogate is plugged into ¨DIG/Sonic 1¨ of your LabPro.
5. Start Loggerpro 3.4.
6. Double-click on the file ¨Pendulum¨ in T:/Ortman-Link/Loggerpro folder.
7. Pull the pendulum away from the photogate. Be sure to displace the pendulum through only a small angle (less than 30 degrees). The theory of this experiment does not apply to larger swing angles.
8. Click on the “Collect” button, release the pendulum, and allow the computer to record the times for about 10 cycles of motion. If the bob hits the photogate, start over again.
9. After about 10 cycles have occurred, click on the ¨Stop¨ button.
10. Using the left mouse button, click and drag the cursor over all of the data points displayed on your graph. The corresponding points in your data table should now be shaded gray.
11. Click on the ¨Statistics¨ button . The average period of oscillation should be displayed as the ¨Mean¨ in the floating box which appears. Record the average in your data table.
12. Select ¨Clear All Data¨ from the ¨Data¨ menu (and click on ¨yes¨) after you have recorded the average period.
13. Repeat steps 7 through 12 for each of the other pendulum lengths listed in your data table.
14. When you are completely finished collecting time data, select ¨Exit¨ from the ¨File¨ menu (do not save your changes.)
15. Open Loggerpro 3.4 again. (Start menu – Vernier Software folder).
16. You will now determine what function of a pendulum's length varies directly with its period. If there is a simple direct relationship between period and length, plotting a graph of period (vertical axis) versus length (horizontal axis) will result in the formation of a straight line. However, if another function of length, such as (length) ½, (length)2, or (length)3 varies directly with the period, a curve will be formed instead. It is very difficult to tell exactly which function it is simply by looking at the curve. Instead, it is helpful to plot a graph of period (vertical axis) versus the function of length in question (horizontal axis). For instance, plotting a graph of period versus (length)2 will result in a straight line only if (length)2 is the correct function. Thus, by plotting period versus different functions of length, the correct function can be found by determining which one yields a straight line.
17. Enter your period (vertical axis) and length (horizontal axis) data. Your data will not be quite a straight line.
18. Select "New Column", "Calculated" from the "Data" menu. Type in "Length Squared" for the New Column Name and "cm^2" for the New Column Units. Click on the space for Equation. Then click on "Variables (Columns)" and select "Length". Finally, type "^2" to square the length.. The formula "Length"^2 should now appear in the Equation space. If this is correct, click on OK.
19. Repeat step 18 for (length)3 and (length)½ functions. To enter the formula for (length)3, use a combination of the "^" and "3" buttons on the calculator after selecting the length column. To enter the formula for (length)½, use the "sqrt" button followed by the length column. Don’t forget to name your columns correctly!
20. A graph of period versus length is currently displayed on your screen. To see whether or not the points form a straight line, click and drag from your first point to your last. Then select "Linear Fit" from the "Analysis" menu.
21. Now you will see whether or not any of the other functions of length form better straight lines. To do this, double-click on the graph and select "More X-Axis Options." Click on the function of length that you would like to check. Then click on OK (twice) to see the graph.
22. Repeat step 21 for each of the functions of length. The one which forms the best straight line is the function which varies directly with the period. You can tell the best straight line by looking at the correlation value. The closer a correlation is to 1.0000, the more of a straight line it is.
22 b. Once you have found this function, make sure that its graph is currently displayed on the screen. You will now use this graph to predict what length pendulum will have a period of 2.0 s. To do this, make sure that this period appears on the plotted graph. If it does not, you will have to rescale the axes by clicking on the arrows above and below the vertical axis label, and to the left and right of the horizontal axis label. Then select "Interpolate" from the "Analysis" menu, and drag the mouse across the graph until you reach a period of 2.0 s. Record the corresponding value for the horizontal axis on your data table. Print your graph by clicking on the printer icon
23. Knowing the function of length which was plotted, use the value recorded in step 22 to determine the length of the pendulum needed to yield a period of 2.0 s.
24. Using a pendulum of the length you calculated in step 23, use Logger Pro (as used previously) to measure its period. Record the actual period in your data table and compute the % error.
THIS IS A FORMAL WRITE UP: You need to include the following:
- A cover page with your name, period, lab partners, date, and title on it.
- Objectives
- Data (typed) and Data Treatment (can be handwritten)
- The graph printed out of the correct value
- Discussion questions – answers typed and in sentence form
- Analysis of Error - % error and at least two sources of error
- Conclusion – at least one paragraph with what you have learned and any improvements you could make to the lab
Data:
Length (cm) / Period (s)75
70
65
60
55
50
45
40
35
30
25
Horizontal Axis Value for Period of 2.0 s: ______
Corresponding Length for Period of 2.0 s: ______cm
Measured Period for Predicted Length: ______s
% Error: ______
Interpretations:
1. According to your graphs, what function of length varies directly with the period of a pendulum? How do you know that?
2. How did the period for your predicted length compare with what it was supposed to be (2.0 s)? What was the % error?
3. The theoretical equation for the period of a pendulum is , where T is the period, L is the length, and g is the acceleration due to gravity. According to this equation, what should be the length of a pendulum whose period is 2.0 s? How does this value compare with your prediction from the graph? How does this value compare with your measured value?
4. According to the equation in #3, would your measured periods be bigger or smaller if this experiment were performed on the moon? Explain WHY!
5. Suppose your grandfather's clock is losing time (running too slowly). Explain what you could do to the pendulum to correct the problem, and explain why it works.
Analysis of Error List your calculated percent errors. State two sources of error that are inherent in the lab – not bad calculations or human error. Please be specific! Explain how these sources of error would affect your data.