Part A
Bungee cords designed to U.S. Military specifications (DoD standard MIL-C-5651D, available at http://dodssp.daps.dla.mil
) are characterized by a force constant in the range of about 10-20 N/m. Suppose that you are designing a bungee jump off of a bridge that is 30.0 m above the surface of a river running below. To get an idea for the maximum cord length, calculate the unstretched length of cord with a spring constant of 10.0 N/m that will result in a 118 kg jumper who leaves the bridge from rest ending their jump 2.0 m above the water's surface. (The 2.0 m builds in a safety margin for the height of the jumper, so you can neglect the height in your calculation.) Since you are finding a maximum cord length, ignore any losses due to air resistance or dissipation in the cord. Ignore the mass of the rope.
System: The jumper (treated as a point particle) plus the earth and the bungee cord. The system constituents interact via gravity, which contributes [gravitational potential energy], and via the restoring force of the cord, which contributes [elastic potential energy]. External influences are assumed negligible.
Model: [Constant Mechanical Energy].
Approach:
Shown above is an initial-state final-state diagram for this situation, along with corresponding energy bar graphs. As indicated in the picture, we have chosen the zero point of the height to be the river's surface. With these pictures in mind, we can set up the Law of Change for our model:
\begin
[ M_
= mgh_
= M_
= mgh_
+ \frac
kx_
^
] \end
Bungee cords provide a restoring force when stretched, but offer no resistance when "compressed", since they fold like an ordinary rope. Thus, the initial spring energy is zero in this case.