Finding and Modeling Effects of Mass, Length, and Release Angle on the Period of a Pendulum

Eiben, et al. |1

Finding and Modeling the Effects of Mass, Length, and Release Angle on the Period of a Pendulum for Small Swings Experiments performed on August 27 t h , September 3 rd & 10 t h , 2018 by Aaron Eiben, John Blasing, and Henry Leach

1 Experimental Design Template Experiment

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2

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Research Question What impacts the period of a pendulum for small swings? [1] Dependent Variable Period Independent Variable Mass Length (68.5 cm),

Control Variables Release Angle (5°) A pendulum’s period is Hypothesis affected by its mass.

The period changes as

Prediction the mass changes.

⋯

Length (from the pivot point to the center of mass)

Mass (50.15 g), Release Angle (5°) A pendulum’s period is affected by its length.

The period changes as the length changes.

⋯ Release Angle Mass (50.15 g), Length (68.5 cm)

A pendulum’s period is affected by its release angle.

The period changes as the release angle changes.

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2 Data Tables and Graphs Table 2-1: Data for testing the effect of mass

Mass (±0.05g) * 50.15 109.9 149.8 189.9 249.7 Average:

Period of Ten Swings (±0.08s) †

Period of One Swing (±0.008s)

16.64 16.59 16.63 16.72 16.76 16.67

1.664 1.659 1.663 1.672 1.676 1.667

Table 2-2: Data for testing the effect of length

Length (±0.25cm) ‡ 11.5 68.5 100.0 150.3 184.0 Average:

Period of Ten Swings (±0.08s)

Period of One Swing (±0.008s)

6.76 16.64 20.12 24.44 27.10 19.01

0.676 1.664 2.012 2.444 2.710 1.901

Table 2-3: Data for testing the effect of release angle

Release Angle (±1°) § 1.0 2.5 5.0 7.5 10.0 Average:

Period of Ten Swings (±0.08s)

Period of One Swing (±0.008s)

16.63 16.71 16.60 16.58 16.59 16.62

1.663 1.671 1.660 1.658 1.659 1.662

Mass uncertainty estimated as scale uncertainty of measuring device.

Period uncertainty estimated by the standard deviation of ten repeated measurements. ‡ Length uncertainty estimated by observer confidence. § Angle uncertainty estimated by observer confidence.

See section 3.3 Uncertainty and Error Analysis for details. * †

Eiben, et al. |3 Table 2-4: Data for testing the effect of length, measuring periods with a photogate

Length (Âą0.25cm) * 25 50 100 150 200 Average:

Period of One Swing (s)

Measurement Uncertainty (s) â€

0.999 1.414 1.997 2.445 2.825 1.936

0.001 0.001 0.001 0.002 0.003

All error bars set to 2đ?œŽđ?œŽ

Figure 2-1: Graph of mass vs. period data ‥

Mass vs. Period 1.70

Period (s)

1.69 1.68 1.67

Average

1.66 1.65 1.64

0

50

100

150

Mass (g)

Length uncertainty estimated by observer confidence.

Period uncertainty estimated by observer confidence. ‥ Mass error bars are too small to be seen at this scale.

See section 3.3 Uncertainty and Error Analysis for details. * â€

200

250

300

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Figure 2-2: Graph of length vs. period data *

Length vs. Period

y = 0.1994x0.5009 R² = 0.9999

50

150

3.5 3.0

Period (s)

2.5 2.0

Average

1.5 1.0 0.5 0.0

0

100

200

Length (cm)

Figure 2-3: Graph of release angle vs. period data

Angle vs. Period 1.69

Period (s)

1.68 1.67

Average

1.66 1.65 1.64 1.63

0

2

4

6

Angle (°)

*

Length and period error bars are small but visible at these scales.

8

10

12

Eiben, et al. |5 Figure 2-4: Graph of length vs. photogate period data *

Length vs Period

y = 0.2001x0.4996 R² = 1

3.5 3.0

Period (s)

2.5 2.0

Average

1.5 1.0 0.5 0.0

0

50

100

150

200

250

Length (cm)

3 Discussion and Conclusion Three experiments are performed to address the question of what impacts the period of a pendulum for small swings. [1] As shown in Figure 3-1, the experimental apparatus is a slotted mass holder hung from a string suspended from a pendulum hanger which allows adjustment of the string length. The system’s total mass is adjustable by changing the number of slotted masses on the holder.

Measurements are made using a triple beam balance, a measuring tape, a protractor, and a stopwatch. As a follow-up to Experiment 2, where length is the independent variable, the experiment repeats with a photogate being used to measure the period of one swing.

Figure 3-1: Experimental Apparatus

*

Length and period error bars are small but visible at these scales.

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3.1 Experiments 3.1.1 Testing the effect of mass The first hypothesis states that a pendulum’s mass affects its period. Therefore, one predicts that the pendulum’s period changes as its mass changes. However, Figure 2-1 does not support this claim. As shown by each point’s error bars, all period data corresponding to different masses lies within two standard deviations of each other and their average value. Following the equivalency criterion established in [2], it is, therefore, possible that these data all represent the same period value and that any observed variation is due to random measurement error. The small range of this uncertainty serves to refute the hypothesis that a pendulum’s mass affects its period. Alternatively, any effect is negligible at this scale.

3.1.2 Testing the effect of length The second hypothesis states that a pendulum’s length affects its period. Therefore, one predicts that the pendulum’s period changes as its length changes. Figures 2-2 and 2-4 strongly support this claim. The periods corresponding to different lengths are so widely separated that their error bars are not visible, let alone overlapping. Thus, by the equivalency criterion, it is highly unlikely that these data represent the same value. Because measurement uncertainty alone cannot easily explain this variation, the hypothesis that a pendulum’s length affects its period remains a viable option.

3.1.3 Testing the effect of release angle The third hypothesis states that a pendulum’s release angle affects its period. Therefore, one predicts that the pendulum’s period changes as its release angle changes. Like the situation of the pendulum’s mass, Figure 2-3 does not support this claim. Again, as shown by each data point’s error bars, all periods corresponding to different release angles lie within two standard deviations of each other and their average value. Given these small uncertainties and the equivalency criterion, it is again not possible to say with confidence that these values are distinct, as all observed variation is explainable by random measurement error. If such a relationship between a pendulum’s period and release angle does exist, its effect must be minimal within the “small swings� domain.

3.2 Comparison with Theory Classical mechanics predicts that a pendulum of length đ??żđ??ż from its pivot point to its center of mass swinging along a small swing arc in a gravitational field đ?‘”đ?‘” behaves like a simple harmonic oscillator with a period đ?‘‡đ?‘‡:

Eiben, et al. |7 đ??żđ??ż đ?‘‡đ?‘‡ = 2đ?œ‹đ?œ‹ďż˝ đ?‘”đ?‘”

Equation 3-1 [3].

One can immediately see that Equation 3-1 does not predict any relationship between the mass or release angle of a small-swing pendulum and its period, which is consistent with the results of Experiments 1 and 3. The equation does, however, show a power relationship between the pendulum’s period and its length, as does Experiment 2. Substituting đ?‘‡đ?‘‡ → đ?‘Śđ?‘Ś s, đ??żđ??ż → đ?‘Ľđ?‘Ľ cm, đ?‘”đ?‘” → 980 cm/s, and converting exact values to decimal numbers, Equation 3-1 becomes đ?‘Śđ?‘Ś = 2.007 đ?‘Ľđ?‘Ľ 0.5

Equation 3-2.

đ?‘Śđ?‘Ś = 0.1994 đ?‘Ľđ?‘Ľ 0.5009

Equation 3-3.

đ?‘Śđ?‘Ś = 0.2001 đ?‘Ľđ?‘Ľ 0.4996

Equation 3-4.

Compare Equation 3-2 with that obtained by fitting the data sets from Experiment 2 to power functions.

For the stopwatch data, that function is

The đ?‘…đ?‘… 2 value of this fit is 0.9999, meaning it accounts for 99.99% of all variation within the experimental data. The percentage error of the fit parameters relative to their theoretical counterparts is 0.65% for the coefficient and 0.18% for the power. For the photogate data, the fit function is The đ?‘…đ?‘… 2 value of this fit is 1, meaning it accounts for 100% of all variation within the experimental data (after rounding, undoubtedly). The percentage error of the fit parameters relative to their theoretical counterparts is 0.30% for the coefficient and 0.08% for the power, less than half those of the stopwatch data. In both cases, these close correspondences between the data and theory provide compelling support for the validity of the theoretical model.

3.3 Uncertainty and Error Analysis 3.3.1 Mass measurement Every measurement of a physical quantity has an associated random error. Because no measuring device is perfectly precise, there will always be some finite range within which the real value of a measured quantity lies. This “scale uncertaintyâ€? is defined as one half of the smallest scale increment [4]. For the triple beam balance used for mass measurements, the smallest scale increment is 0.1g, and so the corresponding scale uncertainty is Âą0.05g. The only way to minimize a scale uncertainty such as this is to use a more precise measuring tool, such as a balance that provides readings down to 0.01g. In that case, the associated scale uncertainty is correspondingly smaller, and random mass measurement errors are likely to fall within that smaller range.

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It is also important to note the technique for determining the mass of a stack of slotted masses which ensures the measurement error falls within the scale uncertainty and not some higher value. Slotted masses should always be added to the mass hanger first, and the mass of the total stack determined with one measurement or the average of a set of repeated measurements. One should not measure each mass individually and add the results together to find the total. Following this method, each measurement has an error within, at a minimum, the scale uncertainty range. The error of the sum then lies within an uncertainty range equal to the sum of the uncertainties of the individual measurements [5]. For example, the mass of a stack of ten slotted masses and one mass hanger, each measured with an uncertainty of ±0.05g, has a total uncertainty of ±0.55g, which is eleven times greater than the measurement uncertainty would be with all masses stacked on the balance and a single measurement of the total mass made. It is also worth pointing out a balance should be adequately “zeroed out” before a mass measurement. Otherwise, it will consistently read higher or lower than the true mass value, which is, therefore, a systematic error. Fortunately, if one identifies such an error, one can subtract it from mass measurements later, but it is still far better to ensure that one uses all instruments correctly from the beginning.

3.3.2 Length measurement Sometimes the scale uncertainty is too small to contain a large percentage of the random errors associated with a measurement, such as when one does not precisely know the position of a point of interest. Measuring the length of the pendulum suffers this condition since this length is the difference in positions of the pendulum’s pivot point and its center of mass, which is not precisely known. The center of mass must, therefore, be estimated. If one assumes that the string’s mass is negligible compared to that of the bob and that the slotted masses have a uniform density, then the center of mass of the pendulum is in the middle of the mass stack. One, therefore, measures all lengths involving mass stacks to the middle of the stack, and in Experiment 1, where the number of slotted masses changes with each data point, the length of the string is changed to keep the distance between the pendulum’s pivot point and the center of its mass stack as constant as possible.

Additionally, the mass hanger itself is not symmetric along its height, and so its center of mass is more difficult to identify. If the mass of the center post is negligible, then the hanger’s center of mass is near the center of its mass disk. However, it appears that the mass of the post is significant enough to offset the center of mass by some amount towards the top of the disk. Therefore, when the empty mass hanger comprises the total mass of the bob, as is the case in Experiments 2 and 3, length measurements are made from the pendulum’s pivot point to the top of the mass hangar disk. The difference between this position and the actual center of mass of the pendulum bob represents a systematic error that could be identified by careful analysis of the mass hanger and then removed from the length data. Alternatively, one can load the mass hangar with every available slotted mass to produce a more uniform mass distribution. Then, the bob’s center of mass will be closer to its geometric center.

To account for all possible length errors described above, a large uncertainty range of ±0.25cm is assumed, and even that might be an underestimate. A careful analysis of the slotted mass system’s mass

Eiben, et al. |9 distribution and a more precise determination of its actual center of mass will allow more precise measurements of the pendulum’s length in the future.

3.3.3 Release angle measurement Another example of when scale uncertainty does not adequately capture random measurement errors occurs when the measurer is not confident in their ability to use the measuring tool as accurately as its precision allows. Such an example within these experiments is the use of the protractor to measure release angles. The protractor’s smallest scale increments, 1°, are closely spaced and can be difficult to distinguish without magnification. Furthermore, the accuracy of the release angle measurement is contingent upon the experimenter’s ability to hold the pendulum at a constant angle between the time one makes the measurement and when the pendulum is released. These factors suggest a more likely angle measurement uncertainty of ±1°, but again that could be an underestimate. Reducing this uncertainty is achievable by using a larger protractor with gradations that are spaced farther apart and therefore more easily determined and building and implementing an apparatus that holds the pendulum at a constant angle until it is released.

3.3.4 Period measurement In these experiments, the experimenter makes period measurements with their eyes and a stopwatch, and the nature of observer response and reaction time introduces the possibility of significant measurement errors. Therefore, a careful analysis is performed to quantify the uncertainty range of these random errors. A pendulum is set up with a specific length and bob mass and released from a specific angle. Its period is measured, and this process is repeated for a total of ten times, all the while maintaining a constant length, mass, and release angle. The standard deviation of these period measurements is computed, and that value is assumed to represent the measurement uncertainty of all future stopwatch period measurements.

Additionally, when determining periods, one measures the time of ten swings, as opposed to the time of one. One then computes the period of a single swing by dividing this measurement by ten. Because the propagated uncertainty of this result is the measurement uncertainty of ten swings divided by the same factor [5], the resulting period uncertainty is a whole order of magnitude smaller. Table 3-1 shows the results of this investigation, which finds that the stopwatch period measurement uncertainty is ±0.008s.

Again, it is crucially important to remember that one achieves such precision by measuring the period of ten swings of the pendulum and computing the period of one swing from that value. Therefore, all stopwatch period measurements must be made using this method if they are to claim this more precise value as the measurement uncertainty.

Of course, another way of reducing measurement uncertainty is to use a more precise measuring instrument, as described in Section 3.3.1. A follow-up to Experiment 2 uses a photogate device to measure periods to the nearest millisecond. The stated uncertainty of this device is ±0.001s [6], but in practice, some period measurements are observed to oscillate around slightly larger values as shown in Table 2-4.

10 | E i b e n , e t a l .

Table 3-1: Data for determining the stopwatch period uncertainty in seconds

Trial

Period of Ten Swings

Scale Uncertainty

Period of One Swing

Scale Uncertainty

1 2 3 4 5 6 7 8 9 10

16.64 16.53 16.45 16.72 16.65 16.71 16.58 16.59 16.58 16.56

0.005 " " " " " " " " "

1.664 1.653 1.645 1.672 1.665 1.671 1.658 1.659 1.658 1.656

0.0005 " " " " " " " " "

Average:

16.60

Measurement Uncertainty

1.660

Measurement Uncertainty

Standard Deviation:

0.08

0.08

0.008

0.008

3.4 Constraints and Generalizability The primary constraint in these experiments is the requirement that the pendulum swings be small, as stated in the research question.

Unfortunately, what “smallâ€? means is somewhat vague. The criterion refers to the fact that pendulums behave as simple harmonic oscillators with periods described by Equation 3-1 only when their swing angles đ?œƒđ?œƒ are small enough that sin đ?œƒđ?œƒ ≈ đ?œƒđ?œƒ, the so-called “smallangle approximation.â€? This range is shown for angles measured in radians in Figure 3-2. When đ?œƒđ?œƒ ≤ 0.2 rad, đ?œƒđ?œƒ and sin đ?œƒđ?œƒ are nearly identical.

This range corresponds to swing angles that are less than 11°. The angle đ?œƒđ?œƒ and sin đ?œƒđ?œƒ are still very Figure 3-2: By Phancy Physicist, from Wikimedia Commons [8] similar up to twice that amount—0.4 rad or about 23°. Beyond three times that range— 0.6 rad or 34°— đ?œƒđ?œƒ and sin đ?œƒđ?œƒ start to diverge considerably and the small-angle approximation breaks down. As this happens, Equation 3-1 alone less and less accurately describes the pendulum’s period, and

E i b e n , e t a l . | 11 because the experimental Equations 3-3 and 3-4 closely match Equation 3-1, they too cannot be generalized to pendulums with large release angles.

More extensive analysis shows that for larger release angles đ?œƒđ?œƒ0 , a pendulum’s period becomes release angle dependent: đ??żđ??ż 1 11 4 đ?‘‡đ?‘‡ = 2đ?œ‹đ?œ‹ďż˝ ďż˝1 + đ?œƒđ?œƒ02 + đ?œƒđ?œƒ + â‹Ż ďż˝ đ?‘”đ?‘” 16 3072 0

Equation 3-5 [7].

Equation 3-5 seemingly contradicts the results of Experiment 3, which does not show a relationship between the release angle of a pendulum and its period. However, that experiment is performed well within the “small swingâ€? regime, with a release angle of 5° or 0.09 rad. Subsequently, the higher order đ?œƒđ?œƒ0 terms become vanishingly small, certainly smaller than any variation caused by random measurement error, and therefore too small to show any effect. Nonetheless, one can in principle test this more general theory using the experimental methods outlined in this report.

4 References [1] University of Cincinnati PHYS 1051L, "Lab 01: Introduction to Experimental Design," [Online].

[Accessed 28 August 2018].

[2] University of Cincinnati PHYS 1051L, "Lab 02: Introduction to Error Analysis," [Online]. [Accessed 4 September 2018].

[3] R. D. Knight, B. Jones and F. Stuart, "Pendulum Motion," in College Physics: A Strategic Approach, 3rd ed., Boston, Pearson Education, Inc., 2015, p. 453.

[4] University of Cincinnati PHYS 1051L, "Pre-Lab 02," [Online]. [Accessed 3 September 2018].

[5] University of Cincinnati PHYS 1051L, "Pre-Lab Lab 04," [Online]. [Accessed 4 September 2018].

[6] University of Cincinnati PHYS 1051L, "Lab 03: Creating Mathematical Models," [Online]. [Accessed 11 September 2018].

[7] R. A. Nelson and M. G. Olsson, "The pendulum – Rich physics from a simple system," American Journal of Physics, vol. 54, no. 2, p. 112–121, February 1986.

[8] P. Physicist, "Wikimedia Commons," 20 May 2011. [Online]. Available: https://commons.wikimedia.org/wiki/File:Small_angle_compair_odd.svg. [Accessed 13 September 2018].