Question

A bungee jumper with mass 64.5 kg jumps from a high bridge. After arriving at his lowest point, he oscillates up and down, reaching a low point seven more times in 44.0 s . He finally comes to rest 20.5 m below the level of the bridge.

Part A

Estimate the spring stiffness constant of the bungee cord assuming SHM.

Part B

Estimate the unstretched length of the bungee cord assuming SHM.

Answer 1

*The bungee jumper has a mass (m) of 64.5 kg
*He hits a low point eight times after his initial in 44 seconds.
*His resting point, when he stops oscillating, is 20.5 meters below the level of the bridge where he began, so this is the equilibrium point of the spring with the jumper attached to it.

Part A)

we need to find the spring constant of the bungee cord, and this is easily found through the equation:
T = (2π)*sqrt(m/k)
Where T is the period of the oscillation, m is the mass of the spring, and k is the spring constant.
So, we have the mass of the spring, which is just 64.5 kg, because the mass of the bungee cord itself is negligible in this situation. But we need to find the period T before we can solve for k.

The period of the spring is how long it takes an object to go from peak to peak in an oscillation. And since we know that the jumper peaked 7 times in 44 seconds, we can simply divide 44 s by 7 to figure out about how long it takes the jumper to go from low-point to low-point. So:
(44 s)/(7) = 6.285 s = T

Now we can take all of out information and solve for k:
T = (2π)*sqrt(m/k)
(6.285 s) = (2π)*sqrt((64.5 kg)/k)
Dividing both sides by (2π) and then squaring both sides gives:
((6.285 s)/(2π))^2 = (sqrt((64.5 kg)/k))^2
(1.0015 s^2/m^2) = (64.5 kg)/k
k = 64.39 N/m = spring constant of bungee cord

Part B)

Now that we have our spring constant, we can easily find the unstretched length of the bungee cord.
To do this, the stretch of the cord can be solved by setting the force of the jumper pulling down on the cord equal to the force of the spring pulling up on the jumper. So:
mg = kΔd
Where m is the mass of the jumper, g is the gravitational acceleration, k is the spring constant, and Δd is the change in distance of the bungee cord (with vs without the jumper attached to it)
Plugging in what we know:
(64.5 kg)(9.81 m/s^2) = (64.39 N/m)(Δd)
Simplifying and dividing both side by (64.39 N/m) gives:
9.826 m = Δd
Since this is the change in equilibrium point with vs. without the jumper, we can simply subtract this value from 20.5 meters, which is the equilibrium point the cord experiences when the jumper is still attached to it. So:
(20.5 m) - (9.826 m) = 10.674 m
This is the length of the unstretched bungee cord.