Lab Report 2016 - 1
Lab Report Outline—the Bones of the Story
Your name and your lab partner(s): Joe Carmody John Carmody Section: 05 Date: Monday November 28
TITLE: Determining the relationship between dynamic equilibrium length of a bungee cord and cord length to model the bungee drop using the CWE theorem.
ABSTRACT:
The purpose of this experiment was to model the relationship between the dynamic equilibrium length of a bungee cord andcord length. The expectations were that the dynamic equilibrium length would increase as the cord length increased because previous experiments had shown that an increase in cord length caused a decrease in the “k” value of the bungee cord. The dynamic equilibrium positions were obtainedby dropping a constant mass from a set height (attached to a set cord length) and measuring dynamic equilibrium position at the bottom of the drop. This was repeated at three different cord lengths to produce a mathematical relationship between the cord length and the dynamic equilibrium length. The mathematical relationship was D= 1.48(C)+0.03 (where D= dynamic equilibrium length and C=cord length). The results agreed with the original hypothesis because the dynamic equilibrium length increased as cord length increased. Potential sources of error were the dynamic equilibrium measurements, the cord length measurements, the constancy of the height at which the mass was dropped, the measurement of the mass, and the fatigue of the bungee cord after stretching. This relationship between cord length and dynamic equilibrium length will allow the modeling of the egg drop using the CWE theorem, which can help predict what cord lengthis required to allow the egg to be as close to the ground as possible.
INTRODUCTION:
The purpose of this experiment was to empirically derive a relationship between dynamic equilibrium length and bungee cord length. Amathematical definition of this relationship (along with the relationship between the K value and cord length already empirically determined in Bungee I) will allow the use of the Classic Work Energy Theorem (CWE) to predict what cord length of bungee cord that will allow the egg to come as close to the ground as possible without hitting the ground in the bungee challenge. According to the Bungee one experiment,
Equation 1
K=(2.294)C-0.968
whereC is the cord length and K is the K-value of the bungee cord at a given equilibrium length. The other relevant equations of this experiment are PEspring=½kx2, PEgravitational=mgh, and (PE+KE)top=(PE+KE)bottom. Using the CWE theorem one can come up with
Equation 2
½kx2=mgh
By substituting equation one and rewriting the variables x and h in terms of dynamic equilibrium length and cord length(x=D-C and h=D), this can be rewritten as
Equation 3
½(2.294C-0.968 )(D-C)2=mg(D)
whereD is dynamic equilibrium length and C is cord length, m is the mass of the object and g is the acceleration due to gravity. If a mathematical relationship between C and D can be experimentally determined, the equation can be used to predict the cord length that will get the egg as close to the ground as possible without making contact. The expectations for this experiment were that the dynamic equilibrium length would increase as the cord length increased because the K value of the bungee cord decreases as the bungee length increases
METHODS:
A constant mass was dropped from a constant height while attached to three different cord lengths. At each cord length, the dynamic equilibrium length was measure three times using a slow motion camera and a reference tape measurer. These measurements were complied to produce a mathematical relationship between the cord length and the dynamic equilibrium length.
Figure 1Experimental Setup: Depicts the moment at which the downward velocity of the mass has reached zero. This moment was captured using a slow motion camera. The cord lengths below show the length of the bungee with no mass attached. The tape measurer was used to measure dynamic equilibrium length. The mass and drop height was constant throughout each trial. Process was repeated 3 times for each cord length.
Drop Height
Tape Measurer
Cord Length (C)
1. 0.2680m2. 0.3750m 3.0.6100m
Mass=0.02500kg (m) (acceleration due to gravity (g) assumed to be 9.81m/s2)
Describesetup:
A mass of 0.02500kg was measured using an electronic scale. A cord length of 0.2680m was measured and tied to the horizontal rod and the mass. Thetop of the mass (now tied to the bungee cord) was held parallel to the top of the horizontal bar, with the center of the mass directly over the point of attachment between the bungee cord and the horizontal bar.The mass was released, the fall was recorded using a slow motion camera, and the dynamic equilibrium length was obtained by reviewing the video and determining the lowest point that the top of mass fell. The camera was held level to the predicted dynamic equilibrium length for more accurate measurements. This process was repeated two more times at this length. The process was then carried out three times for the other two cord lengths.
RESULTS:.
The data collected were dynamic equilibrium lengths that occurred using different cord lengths and constant applied weight. The cord length and dynamic equilibrium position could then be mathematically related, which would help model the bungee mass system using the CWE theorem.
Table 1: Raw Data including cord length measurements, dynamic equilibrium measurements, the dynamic equilibrium length averages for the three trials, and the standard deviation of the dynamic equilibrium lengths.
Dynamic Equilibrium length (m) (±0.007)Cord Length (m) (±0.002) / Trial 1 / Trial 2 / Trial 3 / Dynamic Equilibrium Ave (m) / Std Dev.
0.2680 / 0.430 / 0.436 / 0.444 / 0.437 / 0.007
0.3750 / 0.576 / 0.575 / 0.588 / 0.580 / 0.007
0.6100 / 0.938 / 0.937 / 0.941 / 0.938 / 0.002
The error is greater for the dynamic equilibrium length because the tape measurer was harder to read on the slow motion camera than it was to read directly from the tape measurer. The table shows a consistent increase in the dynamic equilibrium length as the cord length increases as predicted.
Graph 1: The dynamic equilibrium length vs cord length of the bungee mass system. The Equation D=1.48C+0.03 (where D is dynamic equilibrium length and C is the cord length) represents the mathematical relationship between the dynamic equilibrium length and cord length. The slope (1.48) represents the rate at which the dynamic equilibrium length changes with respect to the cord length. The percent uncertainty for the equation is 4.4%.
uncertainty for slope=0.05%uncert= 3.38%
uncertainty for y-intercept= 0.02%uncert= 66.66%
The experimental value of interest is the equation of the linear graph depicted in Graph 1. The slope (1.48) is the rate at which the dynamic equilibrium length increases as the cord length increases. The y intercept of the linear function in conjunction with the slope can be used to predict the dynamic equilibrium length of a bungee mass system. These values were obtained by measuring the dynamic equilibrium position of the bungee cord at different cord lengths and examining the change.
value obtained = 1.48C+0.03
uncertainty=Slope: 0.05 and Y int: 0.02%uncert=Slope: 3.38% and Y-int: 66.66%
Regression Analysis was the method used to calculate the % uncertainty of both these values.
This equation of interest can be used to predict the maximum length a bungee cord will stretch given a constant mass and a cord length using the CWE theorem.
Equation 3
½(2.294C-0.968 )(D-C)2=mg(D)
This predictive model could be a useful asset for success in the bungee challenge. If the dynamic equilibrium length can be predicted from the cord length, we can ensure the egg falls as close to the ground as possible without actually touching the ground.
DISCUSSION:
In order to test the accuracy and precision of the predictive model, the empirically determined dynamic equilibrium lengths were plugged into equation 3 and the cord length was solved for. These cord length values were compared to the measured cord length values from the experiment.
Table 2: The calculated cord length using the measured dynamic equilibrium position and the equation 3 compared to the empirically measured cord length of the bungee cord.
Cord Length (m) (± 0.002m) / Dynamic equilibrium length (m) (± (0.007m) / Calculated cord length (m) (± 0.07m) / % error0.2680 / 0.437 / 0.261 / 2.6%
0.3750 / 0.579 / 0.350 / 6.7%
0.6100 / 0.938 / 0.571 / 6.5%
The average percent error of the calculated values was 5.3%. The percent uncertainty for the equation was 4.4% (calculated using the avoidance method). Because the % uncertainty was less than the percent error, the results indicate our results were not accurate within the experimental uncertainty.
Sources of uncertainty
Potential sources of error were the dynamic equilibrium measurements, the cord length measurements, the constancy of the height at which the mass was dropped, the measurement of the mass, and the fatigue of the bungee cord after stretching. The fatigue of the cord and the measurement of the dynamic equilibrium length were most likely the greatest sources of uncertainty. The slow motion video made it hard to read the measurement values on the tape measurer and the fatigue on the string was noticeable after subsequent drops. The consistency of the height of the drop, the measurement of the cord length, and the measurement of the mass were less significant sources of uncertainty because they were easier to control/measure. One way to decrease the error of the experiment would be to cut the bungee cord into smaller segments, so the effects of fatigue would be reduced.
The conclusions supported our hypothesis. The dynamic equilibrium length increased as the cord length increased. The predictive model was not accurate within the experimental uncertainty, but can still be useful because the percent error and percent uncertainty were within 1%. If this could be accounted for during the bungee challenge, the predictive model could still be useful. One way we could account for the larger percent error would be to subtract 5% from the calculated value of L. This would decrease the likelihood that the egg would hit the ground, but still provide a good dynamic equilibrium range. The better the dynamic equilibrium range, the more likely that the egg will come as close to the ground as possible without crashing.
CONCLUSION:.
Although the predictive model formulated in this experiment was not accurate within the experimental uncertainty, valuable insight was obtain in the experiment. The hypothesis that dynamic equilibrium length increases with cord length was supported. The predictive model generated could be modified to account for the high experimental uncertainty by subtracting 5% from the calculated cord length. This would increase the chances that the egg does not hit the ground during the bungee challenge. This experiment supported the idea of modeling of the dynamic bungee using the CWE theorem. With more careful measurement and improved methodology, a more accurate predictive model for the dynamic equilibrium length can be formulated. The next step in developing a more applicable model would be to measure the deceleration of the mass at the point of dynamic equilibrium. This would elucidate whether the deceleration applied by the bungee cord would shatter the egg (whether it is greater than 3 times the acceleration due to gravity). This information would be helpful in creating an optimal bungee experience.
On my honor, I have neither given nor received any unacknowledged aid on this assignment.
Pledged: Joe Carmody