Baltimore Polytechnic Institute
PENDULUM LABORATORY
Objectives:
1. To prove that the relationship between the period of oscillation, T, and the length, l, of a simple pendulum (see Figure 1) is of the form
– T is proportional to the square root of l.
2. To deduce the value of the constant, C, in the equation for the period of oscillation, T, of a simple pendulum as found in class to be of the following form. In effect, we will prove that the constant is equal to 2p.
3. To observe the null effect of the mass, m, on the period of oscillation.
4. To observe the effect of varying the amplitude on the period (for small angles, the amplitude effect is null).
5. To practice proper laboratory note-taking and reporting.
6. To obtain skill in manipulating and presenting data using Excel.
Focus:
In this lab we are interested in deducing the form of the period equation for simple pendulum motion. Although we derived the form of this equation in class using dimensional reasoning, we will experimentally prove this form and also deduce the unknown constant, which dimensional reasoning does not allow us to do. In addition, we will prove that the equation derived by dimensional reasoning is correct and physically reproducible.
Background:
Although use of pendulums in mechanisms dates back more than 2,000 years, Galileo Galilei is the first person known to have studied pendulum properties around 1600.2 Christiaan Huygens, around 1650, utilized the pendulum in clockwork and provided the world’s first accurate measure of time.3 Among other things,
“[t]he pendulum was crucial for…establishing the collision laws, the conservation laws, the value of the acceleration due to gravity g, ascertaining the variation in g from equatorial to polar regions and hence discovering the oblate shape of the earth, and, perhaps most importantly, it provided the crucial evidence for Newton's synthesis of terrestrial and celestial mechanics.”3
Using the power-law expression and dimensional reasoning as covered in class, the equation for the period of oscillation, T, of a simple pendulum of small amplitude Q (shown in Figure 1 above) is found to be
Equation 1
where l represents the length of the pendulum and g represents the acceleration due to gravity (Note: the “period of oscillation, T,” is the time it takes the pendulum to complete one cycle of motion from the initial starting point to the opposite apex to the initial starting point).
As also discussed in class, the constant, C, cannot be determined by dimensional analysis. The two methods to obtain the constant are experimentation and geometric analysis. The following procedures outline the steps necessary to deduce the value of the constant for simple pendulum oscillation. However, this lab not only describes the procedure necessary to obtain the constant, but the procedure necessary to prove that the mass m of the pendulum does not affect the period of oscillation, as well as the procedure used to prove that the relationship between T and l is indeed of the form
Equation 2
Materials and Setup:
The materials for this lab consist of the following:
1. A stable structure greater than 1m in height from which to hang the pendulum
2. A near-frictionless string/rope (fishing line)
3. Easily interchangeable masses to be hung from the end of the string (e.g., washers or nuts)
4. Reliable instruments for measuring the following:
a. Mass of object(s) hung
b. Length of string
c. Amplitude angle
d. Time length of period
5. Engineer’s notepad and a pen (pen is preferred for its permanence)
The setup for this lab consists of the following:
1. Two students pair to complete the lab together; one student reliably performs actions of pendulum motion, one student documents, times, and instructs/manages the other
2. Setup apparatus
3. Prepare notepad for data collection
4. TEST actions, apparatus, and measurements. Verify accuracy of system
The data table for this lab should contain the following columns:
|Amplitude |Mass |Length |Period |
|(deg.) |(g) |(m) |(s) |
| | | | |
Procedure
In order to perform the appropriate data analysis to fulfill laboratory objectives, the above table must be completed by recording data for numerous trials of the following:
1. Keep Length and Amplitude constant while varying the Mass
a. Set the length of the pendulum to 1.0m.
b. Begin with amplitude of 10o and your smallest mass.
c. Record the period for 3 trials with your smallest mass:
i. Start timing when the mass is released from the amplitude.
ii. Stop timing when exactly 5 periods have been completed. You will take this time and divide it by 5 to obtain the average period for that trial. Repeat this process 2 times to obtain a total of 3 trials for Amplitude = 10o, Mass = smallest, Length = 1.0m
d. Repeat this procedure for the remaining two masses, keeping length and amplitude constant at 1.0m and 10o.
2. Keep Length and Mass constant while varying the Amplitude
a. Set the length of the pendulum to 1.0m.
b. Begin with amplitude of 5o and your heaviest mass.
c. Start timing when the mass is released from that amplitude.
d. Stop timing when exactly 5 periods have been completed. You will take this time and divide it by 5 to obtain the average period for that trial. Repeat this process 2 times to obtain a total of 3 trials for Amplitude = 5o, Mass = heaviest, Length = 0.1m.
e. Repeat this procedure for the following amplitudes: 10o, 15o, 20o
f. Then, repeat the procedure 1.a-f for the lengths 0.8m, 0.6m, 0.4m, 0.2m.
Data Analysis
Objective 1: To prove that the relationship between the period of oscillation, T, and the length, l, of a simple pendulum (see Figure 1) is of the form – T is proportional to the square root of l.
For this exercise, forget that we’ve already derived Equation 1 above; all we have is the raw data recorded during the experiment above. How, then, could we deduce the relationship between T and l? This analysis is outlined below:
1. PLOT 1 – Create an x-y scatter plot of T vs. l for the amplitudes 5o and 10o on two separate plots. You should notice a scatter similar to the data in Figure 2 at the right. A first glance at this plot may lead you to presume a linear relationship between T and l. However, with limited data points, we shouldn’t assume a linear relationship.
2. PLOT 2 – We can assume a power relationship between T and l such that Tala; in other words, T is proportional to l to some power a. If a is 1, the relationship between T and l is linear. If a is anything other than 1, the relationship between T and l is not linear. So, we need to graphically determine the relationship since that makes the best use of the data you’ve collected.
a. DERIVATION:
T a la
Log T a a*Log l Equation 3
b. If Equation 3 is accurate, we could create an x-y scatter plot of Log T vs. Log l. This plot will give a straight line. The slope of that line, then, is the power a for which we are searching. Do this and find the power a.
Objective 2: To deduce the value of the constant, C, in Equation 1 above.
For this exercise, forget that the constant has already been deduced by others before us to be equal to 2p. We’ll now begin with basic manipulation of our original equation to put in into a linear form that will enable us to more easily deduce the constant than is possible directly from our raw data.
1. EQUATION MANIPULATION – we’ll start with our basic pendulum equation, Equation 1 above, isolate the constants C and g from our variable l, then utilize this new form in a Log-Log plot to determine the value.
Where Equation 4
Log T = (1/2)*Log l + Log g Equation 5
2. PLOT 3 – You should notice that Equation 5 is in slope-intercept form. Therefore, it follows that a plot of Log T vs. Log l will have a slope equal to ½ and a y-intercept equal to Log g. Produce an x-y scatter plot of Log T vs. Log l. If you show the equation of the line on the plot, the y-intercept is given. Use the value of the y-intercept, Equation 4, and the acceleration due to gravity equal to 9.81 m/s2 (check all units) to determine the value of C. (Hint: if Log b = a, then b = 10a).
Objective 3: To observe the null effect of the mass, m, on the period of oscillation.
For this exercise, plot data accordingly and describe the results.
Objective 4: To observe the effect of varying the amplitude on the period (for small angles, the amplitude effect is null).
For this exercise, plot data accordingly and describe the results.
Lab Write-Up
1. Produce a brief structured report, including well-labeled graphs, which describes the experiment, the method, the results, and the errors.
2. What unexpected or expected difficulties did you encounter in the procedure, and what were your method(s) for extracting accurate data in the face of those difficulties?
3. Using the data you obtained with mass maximized and amplitude 10o, extrapolate the length of string required for the period of oscillation of the pendulum to equal 1 second. What usefulness does this information provide? Explain your process and show all work.
4. Using your raw data for period and length, and assuming that the constant C equals 2p, find a best-value for the acceleration due to gravity. Explain your process and show all work.
5. Be sure to appropriately include any references, including data borrowed from other groups (if applicable). Remember, unreferenced data and unreferenced information is stealing and fraud.
Works Cited
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Figure 2
Plot 1 Sample
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Figure 1
Simple Pendulum1
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