How Far Does Bob Plunge and What is the Maximum Force on the Bungee Cord?

[Snikt]

"An 80 kg person jumps off a bridge with a bungee cord of unstressed length 100 meters and a cross sectional area of 400 mm squared. Assume cord has zero weight, how far into the gorge does Bob plunge and what is the maximum force in the cord? The cord has a modulus of Elasticity E = 8 MPa."

The furthest I can get with this question is solving for mg = kx at the equilibrium point, and the distance from that point to the unstretched 100 meters is around 124 meters. How do I carry forward is beyond me. Can someone please explain to me how I can solve for the distance from the equilibrium point to the bottom of the plunge?

Thanks for your time.

Snikt

 

[Chi Meson]

I don't know if you are at this point, but if you solve using conservation of energy, it would be simpler:

At the farthest stretch of the cord, a certain amount of gravitational potential energy is transformed into elastic potential energy. call "y" the maximum stretch of the cord and "y + 100" the height the jumper has fallen.

 

[M98Ranger]

astic,

To solve for the distance from the equilibrium point to the bottom of the plunge, we can use the equation for the potential energy of a spring: PE = 1/2kx^2, where k is the spring constant and x is the distance from the equilibrium point. In this case, the bungee cord is acting like a spring, so we can use this equation to find the distance.

First, we need to find the spring constant, which is given by the modulus of elasticity E and the cross-sectional area A of the cord. The formula for the spring constant is k = EA/L, where L is the length of the cord. Plugging in the values given in the question, we get k = (8 MPa)(400 mm^2)/(100 m) = 3200 N/m.

Next, we can use the given mass and acceleration due to gravity (g = 9.8 m/s^2) to find the force of gravity acting on the person: Fg = mg = (80 kg)(9.8 m/s^2) = 784 N.

At the equilibrium point, the force of gravity is equal to the force of the bungee cord pulling back up, which is given by the spring force equation: F = kx. Setting these two forces equal to each other, we get 784 N = (3200 N/m)x. Solving for x, we get x = 0.245 m.

This means that at the equilibrium point, the person has fallen 0.245 meters from the unstretched length of the bungee cord. To find the distance from the equilibrium point to the bottom of the plunge, we need to add this distance to the unstretched length of the cord (100 meters). Therefore, the distance from the equilibrium point to the bottom of the plunge is 100.245 meters.

To find the maximum force in the cord, we can use the same equation for the spring force: F = kx. At the bottom of the plunge, the person has fallen a total distance of 100.245 meters, so the maximum force in the cord is given by F = (3200 N/m)(100.245 m) = 320245 N. This is the maximum force that the cord can withstand before breaking.

I hope this explanation helps you understand how to solve for the distance and maximum force in this scenario. Let me know if you have any further