|
(Photo courtesy Wikimedia Commons |
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 times unstretched length in the range kL ~ 800-1500 N. Jumpers using these cords intertwine three to five cords to make a thick rope that is strong enough to withstand the forces of the jump. Suppose that you are designing a bungee jump off of a bridge that is 50.0 m above the surface of a river running below. You have read that you should expect the cord to stretch (at peak extension) to about 210% of its natural length. You have also read that you should use 3 cords together for jumpers with weights in the range 100-150 lbs, 4 cords for 150-200 lbs, and 5 cords for 200-250 lbs. Suppose you decide to use cords of length 20 m, which would seem to offer a safety zone of about 8 m (or really about 6 m if the cord is attached at the ankles).
Part A
Find the expected maximum length of the cords for a 200 lb person jumping with 4 cords, so that kL = (4)(800 N) = 3200 N. Since you are evaluating the safety factor, ignore any losses due to air resistance or dissipation in the cord. Ignore the mass of the rope.
Solution
System: The jumper (treated as a point particle) plus the earth (treated as an object of infinite mass) and the bungee cord and the bridge (also treated as an object of infinite mass).
Interactions: The system constituents interact via gravity, which contributes gravitational potential energy and via the elastic restoring force of the cord which contributes elastic potential energy. External influences are assumed negligible.
Model: [Mechanical Energy and Non-Conservative Work].
Approach:
We begin with an initial-state final-state diagram and the corresponding energy bar diagrams:
|
|
Initial State |
Final State |
|---|
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
[ E_
= mgh_
= E_
= mgh_
+ \frac
k x_
^
] \end
Bungee cords provide an elastic 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.
It seems that we have a problem here, because we do not know hf or xf. The easiest way to deal with this problem is to utilize our freedom to choose the zero point of the height axis. If we restructure our coordinate system to place h = 0 m at the point where the cord is stretched to its natural length L (as shown below).
|
|
Initial State |
Final State |
|---|
This redefinition of the origin of the height in our coordinate system greatly simplifies the equation describing the evolution of the jumper's mechanical energy:
\begin
[ mgL = - mgx_
+ \frac
k x_
^
] \end
You can also solve using the initial coordinate system. You would simply have to substitute hf = 50 m - L - x. After cancelling mg(50 m) from each side, you recover the same expression.
The discussion around the diagrammatic representation has left us with a quadratic equation in xf, which is solved to obtain:
\begin
[ x_
= \frac{mg \pm \sqrt{(mg)^
+ 2 k mg L}}
]\end
It is not sensible that we should find a negative value for xf, so we must select the plus sign, giving:
\begin
[ x_
= \frac
\left(1+\sqrt{1+\frac
{mg}}\right) = 21.5\:
] \end
Does this bear out the original estimate that the cord should stretch to 210% of its initial length?
Can you convince yourself by looking at the form of the symbolic answer that a 150 lb person attached to 3 cords and a 250 lb person attached to 5 cords would have the same xf as the 200 lb person on 4 cords?
Part B
Based on your analysis from Part A, what would be the peak force acting on a 200 lb person making the jump attached to 4 cords?