Bungee Jumper Physics Lab Question

[flouran]

Homework Statement

Hi,
So I have a final lab in my college physics class. And the lab procedure is as follows:

Make or use a bungee cord by tying ten or eleven #19 latex rubber bands end-to-end. Attach the upper end high enough so that when 200 grams (0.2 kg) is hung from its lower end, it will almost touch the floor. Begin with a weight of about 0.2 N (0.0204082 kg) and measure the extension of the bungee cord as a function of the applied force up to a maximum extension of 1 to 12 meters. Also hang the Super Hero on the bungee cord and measure the resulting extension. The purpose is to predict, given a particular bungee cord, the minimum height above the floor necessary to ensure (or insure :razz:) that the jumper comes within 5-10 centimeters (0.05 to 0.1 meters) of the floor.

Note: The measured mass of the Bungee jumper is 0.25837 kilograms (258.37 grams).
I have attached my data as an Excel file:
View attachment AP Physics Post Lab.xls.
My question is, what formula does k follow (it is most definitely not linear)?

Homework Equations

Thus, once I find k, it should hopefully be easy to compute this minimum height using energy considerations (please let me know if I am somehow wrong):
$$U_i + K_i = U_f + K_f$$,
Since the object is released from rest,
$$mg(h_{min}-{L_0}) = mg(0.05)+0.5k(h_{min}-0.05-L_0)^2$$,
where $$h_{min}$$ is the minimum height (the thing I need to calculate), m is the mass of the jumper = 0.25837 kg. $$L_0$$ is the initial length of the bungee which I measured to be 0.395 meters.

The Attempt at a Solution

In the attached Excel file I included a graph of the empirical computation of k (y-axis) versus the attached mass in kilograms (x-axis). As you can see, k asymptotically approaches 4 N/m, but is non-linear. I estimate k to be around 7 N/m if the bungee jumper is attached.

 

[nvn]

flouran: I don't think I would bother with computing k, for the main calculations, although k is informative (for information only).

Your empirical data, although extremely coarse, suggests that a better regression would be P(x) = 17.24*x/(1 + 4.786*x - 1.886*x^2), where P(x) = applied force (N) as a function of x, and x = deflection (m). Thus, U1₁ + K₁ = U1₂ + K₂ + U2₂, where U1 = potential energy, and U2 = strain energy. Therefore,

m*g*h1 + 0 = m*g*h3 + 0 + integral[P(x)*dx],

where h1 = jumper initial height above ground, h3 = jumper final height above ground, and the integral is integrated from 0 to h2 - h3, where h2 = height of unstretched bungee cord lower end above ground (the unknown). Unless you can find an analytic solution to the above integral, which I did not attempt, you could solve the problem numerically, by trial and error, if you wish.

 

[nlightner]

Hello,

Thank you for sharing your lab procedure and data. Bungee jumping is a great way to apply the principles of physics in a real-life scenario.

To answer your question, the formula for k in this case would be related to Hooke's Law, which states that the force exerted by a spring or elastic material is directly proportional to the extension or compression of the material. In this case, the bungee cord can be treated as a spring with a spring constant, k.

Based on your data, it seems that the relationship between k and the attached mass is not linear, which is expected. The bungee cord will behave differently when a heavier mass is attached to it, compared to when a lighter mass is attached. This is because the heavier mass will cause more extension in the bungee cord, leading to a higher value of k.

To calculate the minimum height necessary for the jumper to come within 5-10 centimeters of the floor, you can use the formula you mentioned, which is based on the conservation of energy. However, you would need to take into account the initial potential energy of the bungee cord, which can be calculated using the spring constant, k, and the initial length, L_0. Your final equation would look something like this:

mg(h_{min}-{L_0}) = 0.5k(h_{min}-0.05-L_0)^2 + 0.5kL_0^2

Solving this equation for h_{min} would give you the minimum height required for the jumper to reach within 5-10 centimeters of the floor.

I hope this helps. Good luck with your lab!