Lab Practical: Bungee Jumping Challenge
In this scenario, we have a partially elastic bungee cord with static line length s and elastic line length l. We need to predict the length s required to land the bungee "jumper" safely within a certain range. The total height will be a sum of the distance between the ceiling and platform, x, and the drop height, d. We will know the length l of the elastic cord after we create the cord itself; l will be between 50 and 80 cm due to the amount of available string. On the day of the drop, we will be given the mass m of the "jumper," which will be between 100 and 300 g, and the drop height d, which will be between 160 and 260 cm.
Determining the Spring Constant
The first thing that must be determined is the spring constant k of the elastic string. In order to do this, we use Hooke's law, F=kx. The following experimental procedure was followed.
Attach your cord to the end of a force sensor, as shown at left. Drag the force sensor in even increments of displacement and hold, repeating until you have as many data points as you want.
The data studio graph will look like the image in the middle. At each plateau, measure the force. The displacement will increase by the even interval that you selected at each plateau. |
Hooke's law expresses that F=kx, which follows the format of a linear equation y=mx+b. Thus, using the slope a Force vs. Displacement graph, we can find the spring constant k. The data and graphs are shown below.
According to the graph, the spring constant was determined to be 13.000 N/m with a strong correlation coefficient of 0.980.
Bungee Drop Derivation
In order to accurately predict the length s of the static cord using these variables, we need to derive an equation that includes only known and constant variables, as shown below.
Final Drop
The spring constant was determined to be 13.000 N/m. The length l of elastic cord used from knot to knot was 0.68 m. The drop distance was 2.29 m. The mass was .100 kg. The x distance is already known to be 0.188 m.
Using these values, the predicted length s was calculated to be 1.21 m.
Using these values, the predicted length s was calculated to be 1.21 m.
The mass landed within the 4 point range. The bungee fell too short to reach the desired range.
This error can be attributed to Hooke's Law. When we compared Hooke's Law to y=mx+b, we assumed that force was directly proportional to displacement. However, the graph is more a curve than it is a perfectly straight line, and the force was probably directly proportional to a polynomial value of x instead. Thus, the k value would not be constant throughout all of the displacements and we would need multiple equations for bigger displacements. If our spring constant would have been more constant, the error would have been less likely to occur.
This error can be attributed to Hooke's Law. When we compared Hooke's Law to y=mx+b, we assumed that force was directly proportional to displacement. However, the graph is more a curve than it is a perfectly straight line, and the force was probably directly proportional to a polynomial value of x instead. Thus, the k value would not be constant throughout all of the displacements and we would need multiple equations for bigger displacements. If our spring constant would have been more constant, the error would have been less likely to occur.