Question

A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 93.0 kg ) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 420 N .

When you do this, what distance will the bungee cord that you should select have stretched?

Answer 1

Solve this problem using work - energy theorem -  

W_ext = ΔK + ΔU + ΔE_th

In the above expression, W_ext is the work done by external forces on a system, ΔK is the change in kinetic energy of objects in the system, ΔU is the change in potential energy due to conservative forces within the system, and ΔE_th is the change in thermal energy due to non-conservative forces within the system.

So we need to choose a system. I'll choose a system that consists of your father-in-law, the Earth, and the bungee cord. With this choice, there are no external forces doing work on the system (W_ext = 0), and there are no internal non-conservative forces transforming mechanical energy to thermal energy (ΔE_th = 0).

0 = ΔK + ΔU + 0

If we assume the bungee cord is massless, then only your Father-in-law and the Earth can have kinetic energy changes. However, the Earth's kinetic energy change will be negligible, so ΔK only need be considered for your FiL. But your FiL is at rest on the bridge, and is also at rest at his point of maximum descent, so ΔK = 0.

0 = 0 + ΔU + 0

There are two conservative forces internal to the system that can cause potential energy changes, the gravitational force and the elastic force due to the bungee cord.

0 = ΔU_g + ΔU_e = [ mg(h_f) - mg(h_i) ] + [ ½k(s_f)² - ½k(s_i)² ]

where i and f denote initial and final states, respectively. Choose zero points. If one wishes to use the formula U_e = ½ks² , then the zero point for elastic potential energy must be when the bungee cord is relaxed, so s_i = 0. I'll choose the zero for potential energy at the point where your FiL has fallen the relaxed length of the bungee cord. If the bungee cord has relaxed length L = 30m and then stretches a distance x = 11m (so your Father-in-law falls a total distance L + x = 41m), this means

s_f = x
h_i = +L
h_f = -x

so

0 = [ mg(-x) - mg(L) ] + [ ½k(x)² - 0 ]

Solve for k.

mg(L + x) = ½k(x)²

k = 2mg(L + x) / x²

Put the values -

k = 2 (93.0 kg) (9.81 m/s²) (30 m + 11 m) / (11 m)²

=> k = 618.3 N/m

Now, a force of 420 N applied to a bungee cord with this k will stretch a distance

Therefore -

d = F/k = (420 N) / (618.3 N/m) = 0.680 m

So, the bungee cord will stretch a length of 'd' = 0.680 m (Answer)