# Tension in a Bungee cord

1. Jun 3, 2009

### sunniexdayzz

1. The problem statement, all variables and given/known data

Consider a bungee cord of unstretched length L0 = 43 m. When the cord is stretched to L > L0 it behaves like a spring and its tension follows the Hooke’s law T = k(L − L0). But unlike a spring, the cord folds instead of becoming compressed when the distance between its ends is less than the unstretched length: For L < L0 the cord has zero tension and zero elastic energy. To test the cord’s reliability, one end is tied to a high bridge (height H = 147 m above the surface of a river) and the other end is tied to a steel ball of weight mg = 120 kg×9.8 m/s2. The ball is dropped off the bridge with zero initial speed. Fortunately, the cord works and the ball stops in the air 14 m above the water — and then the cord pulls it back up. Calculate the cord’s ‘spring’ constant k. For simplicity, neglects the cord’s own weight and inertia as well as the air drag on the ball and the cord. Answer in units of N/m.

2. Relevant equations
F=ma
T=k(L-L0)
gravity = 9.8m/s^2

3. The attempt at a solution
I tried to figure out the Tension by using f=ma
I assumed that acceleration at the bottom of the rope was 0 .. but now thinking about it i don't think that's true. so then if T is tension and G is the mass x gravity then T + G = ma

then i would plug in T for hooke's law and solve for k.

I guess my biggest problem is figuring out what T is.

2. Jun 3, 2009

### Dr.D

This is a dynamics problem, so you better treat it as such.

Consider the motion in two phases.
Phase 1: There is no tension in the cord, the ball has not fallen far enough to pull the cord tight.
Phase 2: The ball is continuing to fall, with the cord acting like a spring.

Solve the first phase, and find out the time when the first phase ends, the position and velocity at that time. These become the initial conditions for the second phase.

Solve the second phase, using the terminal conditions from the first phase as initial conditions. Carry the solution to the max displacement which was a given value. Use this to evaluate your final constants.

Now you should be able to back out the required value for the spring constant.

It is just a little bit more than simple plug and chug!

3. Jun 4, 2009