An Analysis of Initial Vertical Displacement or

Height’s Effects on Height Loss of a Pendulum

Mr. MacIB SL Physics B404 October 2013

DESIGN

Defining the Problem and Selecting Variables

Researchable Question.

How does changing the initial vertical displacement (“height”) of a pendulum from its state of rest affect the height loss after 5 back-and-forth motions (‘swings’) of the pendulum?

Hypothesis.

If the initial height of the pendulum is increased, then the height loss after 5 swings of the pendulum is predicted to linearly increase in magnitude. The reasoning behind this begins with the premise that the amount of potential energy in a system is defined as its mass multiplied by the acceleration due to gravity multiplied in turn by the vertical height of the object. As the same pendulum is used throughout, and the entire experiment is carried out on earth, the mass and acceleration due to gravity will remain constant. Thus increasing the initial height of the pendulum will linearly increase its potential energy (and thus its kinetic energy), but it will also increase the length of the arc it must travel. The arc length increases proportional to the angle’s increase, which increases according to trigonometric relations rather than linear ones. Accounting for air resistance, which has more time to have a greater effect on the pendulum’s path as initial height increases, it was agreed for simplicity’s sake to assume a linear, positive correlation between initial height and height loss after 5 swings.

Variables. Independent: the initial vertical displacement or (specific to this experiment) the

initial “height” of a pendulum’s position from its state of rest. Two meter sticks, placed at both ends of the table that the clamp holding the pendulum is attached to, will act as frames of reference. After taking a third meter stick to measure the “height” of the pendulum at rest, the same meter stick will also be used to position the pendulum for its predetermined vertical displacement from its state of rest.

Dependent: the height loss after 5 back-and-forth motions (or “swings”) of the pendulum. Video analysis of all trials using a program such as Logger Pro (used in this experiment) will be used to determine the loss in height of the pendulum after 5 swings of the pendulum.

Controlling Variables

In order to minimize the chances of other factors affecting the height loss of the pendulum after 5 swings, many controls had to be employed. Several of the more significant ones are described below:

Length of the string attached to the pendulum: To keep this constant, the same string was simply used for all trials. Because length is one of the factors affecting the true period of the equation describing the pendulum’s position over time (as

seen in the equation 2π √Lg

), it is especially important to keep this value the

same. Using the same string length ensured the constancy of this length to within 0.01 m, as will be described later in the Method section.

Acceleration due to gravity: Again using the equation above, one sees that if the acceleration due to gravity is not constant, then a different equation for the pendulum’s position over time arises. Since all trials were performed within a 1 m3 area on Earth, the acceleration due to gravity was assumed constant.

Person releasing the pendulum: Different people have different methods of releasing objects, and even with only one person, it is not feasible to expect that in every single trial the ball will be released the exact same way. However, using only one person to release the pendulum lowers the probability that drop method affects the data. To keep this more constant in future experiments, simply specify in the method for the class to break up into small groups and then have them distribute roles in the lab, including one student per group with the duty of releasing the pendulum.

Person analyzing the pendulum using computer software: Different people will most likely make different judgment calls when deciding on which pixel the origin is at, which pixel the center of the bob (in this experiment, a golf ball) attached to the pendulum is at after five swings, etc. Thus to control this variable simply state in the procedure to have one person who uses the videos of the trials to analyze the height loss of the pendulum from initial conditions to conditions immediately after five swings.

Frame rate in Logger Pro: Having different frame rates for different trials will cause some of the data to be more accurate (and most likely more precise) than the rest. One trial should not have an “advantage” over another except in the form of the randomness of the universe.

Method for Collection of Data

DATA COLLECTION AND PROCESSING

Recording Raw Data

Initial Height Height Loss Trial 1 Height Loss Trial 2 Height Loss Trial 3 Observations

(m ± 0.01 m)(m ± 0.001000 m) (m ± 0.001000 m) (m ± 0.001000 m)

0.05 0.010650 0.005368 0.003231 Did the ball lose height?

0.10 0.006431 -0.002141 0.006471 Redid bad trial0.15 0.002128 0.026740 0.020290 Redid 2 bad

trials0.20 0.043670 0.044740 0.035250 All 3 trials

went smoothly; mostly visible

drop0.25 0.05020 0.065418 0.037430 Drop much

more visible0.30 0.033063 0.071670 0.074010 Redid 1 trial

because of something

strange in the original 2nd

trial

Processing and Presenting Raw Data

Calculations.

Initial Height Initial Height Uncertainty % Uncertainty Initial Height Height Loss Trial 1 Height Loss Trial 2 Height Loss Trial 3 Height Loss Uncertainty0.05 0.01 20.00 0.010650 0.005368 0.003231 0.0010000.10 0.01 10.00 0.006431 -0.002141 0.006471 0.0010000.15 0.01 6.67 0.002128 0.026740 0.020290 0.0010000.20 0.01 5.00 0.043670 0.044740 0.035250 0.0010000.25 0.01 4.00 0.050200 0.065418 0.037430 0.0010000.30 0.01 3.33 0.033063 0.071670 0.074010 0.001000

Sample Calculations.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.000000

0.010000

0.020000

0.030000

0.040000

0.050000

0.060000

0.070000

f(x) = − 12.2604197530865 x³ + 6.75786084656087 x² − 0.836640670194006 x + 0.0325202222222222R² = 0.99054210336952

(Cubic) Initial Height (m) vs. Average Height Loss (m)

Initial Height (m)

AverageHeightLoss(m)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.000000

0.010000

0.020000

0.030000

0.040000

0.050000

0.060000

0.070000

f(x) = 0.321140476190476 x² + 0.134997595238095 x − 0.00610010000000002R² = 0.937718656010609

(Quadratic)Initial Height (m) vs. Average Height Loss (m)

Initial Height (m)

AverageHeightLoss(m)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.000000

0.010000

0.020000

0.030000

0.040000

0.050000

0.060000

0.070000

f(x) = 0.247396761904762 x − 0.0135933777777778R² = 0.929366705546323

(Linear)Initial Height (m) vs. Average Height Loss (m)

Initial Height (m)

AverageHeightLoss(m)

CONCLUSION AND EVALUATION

Concluding Based on Evidence

We can conclude that a relationship most likely exists between initial height and average height loss after 5 swings, though of what kind, we do not know. Out of the simplest regression examples, a cubic relation seems most feasible; only one point with its error bar uncertainty does not lie on the best-fit cubic curve, and the R2 is 0.9905, interpreted to mean that 99.05% of the variation in the average height loss is explained by the proposed cubic relationship with initial height. Meanwhile, when considering the best-fit quadratic relationship between initial height and average height loss after 5 swings, three points (even with their errors bars) fall outside the range of the curve, and its R2 is 0.9377, which, while still quite high, is certainly less than that of the cubic curve (0.9905). Furthermore, for the best-fit linear curve as well, three points (even with their error bars) fall outside the range of the line, and its R2 is 0.9294, which, while still quite high, is certainly less than that of the cubic curve (0.9905) as well as that of the quadratic curve (0.9377).

The two variables in general seem to be in a direct relationship; this is seen primarily from the graphs. As initial height increases from 0.05 m up by 0.05 m through 0.30 m, the average height loss starts at 0.006416 and continues to 0.003587 then 0.016386, afterwards to 0.041220, to 0.051016, and finally up through 0.059581. However, we now notice the outlier in the set of average height loss. With the initial height of 0.10 m, one of the individual trials actually had a gain in height rather than a loss! This implies that either significant method or random error occurred at that point (if it had been equipment error, then why did none of the other trials return a negative result for height loss?) – perhaps the dropper accidentally

transferred kinetic energy to the pendulum; or perhaps the particular, random conditions during that one trial randomly gave the pendulum enough kinetic energy to swing back up past its initial height even after 5 swings.

Another outlier is the average height loss at initial height 0.20 m (0.041220); this point was not on the best-fit curves for any of the three graphs, not even with its error bars. It had an abnormally high difference in magnitude from the previous average height loss (0.016386 at initial height 0.15 m). The differences in magnitude from the previous average height usually range from ± 0.015 m, yet for the average height loss at initial height 0.20 m, the difference in magnitude from the previous average height is over 0.025 m. It is possible that some other variable that was not considered in this experiment affects average height loss after a certain initial height; however, more error in the form of significant method or random error seems more feasible.

One major limitation in this experiment was one of the many calibrations in Logger Pro, in particular its frame rate. Frame rate was kept constant at 30 frames per second, but if the constant frame rate had been increased to, say, 40 frames per second or even greater, the calculations of height loss (and subsequently average height loss) would have been more accurate (and hopefully, as a result, more precise). Since only 30 frames were taken per second, some valuable data may have been cut out from the numerical analysis. An easy fix to this problem is to, in future, specify in the method to customize the frame rate in Logger Pro depending on the type of experiment one is performing; for an experiment such as this one, at least 30 frames per second is advised. A lesser frame rate would cut out even more valuable data and decrease accuracy still more; a greater frame rate, up to a point, would increase the accuracy of the data, something needed to assess the validity of the data.

Another major limitation was the length of string that was chosen for this lab. A length of 0.270 m ± 0.001 m was used during all trials, and while this length was indeed kept constant, if a longer string length had been chosen, then the results most likely would have been more

noticeable. This is due to the fact that the period of a pendulum, given by the equation 2π √Lg

,

where L is the length of the pendulum, would increase as L increased (L is in the numerator of the fraction and not the denominator). An increased period of a pendulum implies that the eye would have more time to see the drop in height of the pendulum after 5 swings. In future, it is advisable to specify in the procedure to have a string length of, say 0.400 m ± 0.001 m. Then the pendulum would be long enough for the naked eye to notice the drop height (especially important in the earlier trials, where initial height is very little) and thus for the experimenters to have greater appreciation for the results in the lab.

However, probably the most significant weakness of this lab by far was that the method for dropping the pendulum was totally unspecified. Not only do different people usually have different dropping methods (decreasing the repeatability of the experiment), but also, since this dropping method was treated in such a cursory manner, it is likely that a number of the data points were thrown off due to differences in dropping method and/or the pendulum being

accidentally pushed and not just dropped. This could potentially explain all of the outliers in the data, such as the abnormally low average height loss at initial height of 0.10 m (and/or the negative, individual data point that caused the abnormally low average); the abnormally high average height loss at initial height of 0.20 m; why in the trial with initial height of 0.30 m an individual data point still has height loss of 0.03 m when that height loss was first reached in the 0.20 m trial; why the trial with initial height of 0.15 m has such a large uncertainty; etc. Thus, the most important advice to be given to those who in the future perform this experiment is to specify in the procedure a clear, concise method that can consistently be used to drop the pendulum.