Conservation of Elastic and Gravitation Energy - 2

[PeachBanana]

Homework Statement

A 70 kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 13 m , and falls a total of 37 m .

Calculate the spring stiffness constant of the bungee cord, assuming Hooke's law applies.

88 N/M

Calculate the maximum acceleration she experiences.

Homework Equations

Spring Force = -k * x
F = m * a

The Attempt at a Solution

Spring Force = (88 N/M)(24 m)

I used 24 m because that was the maximum stretch of the cord.
Spring Force = 2,112 N

a = F / m
a = (2112 N) / (70 kg)
a = 88 m/s^2
I added 9.8 m/s^2 because she was in free fall.
a = 97.8 m/s^2

Could someone explain why this is incorrect?

 

[Delphi51]

a = (2112 N) / (70 kg)
a = 88 m/s^2

I get 30, not 88.
Why add 9.8? My thinking is that
F = ma
kx - mg = ma
a = kx/m - g
Maximum (upward) acceleration is when x is at its maximum but
9.8 would be subtracted rather than added.

 

[PeachBanana]

I made a typo on the acceleration, oops. I added 9.8 m/s^2 because I thought the jumper was moving in the same direction as gravity as she jumped. Could you explain why you subtracted?

 

[Delphi51]

kx - mg = ma
The spring force kx is upward. Gravity mg is downward.

 

[asteriatic]

I would like to clarify that the calculation provided is not entirely incorrect, but it is missing a few key factors. Firstly, the maximum stretch of the cord should be 37 m, not 24 m. This is the total distance the bungee cord stretches from its unstretched length of 13 m. Secondly, Hooke's law states that the spring force is directly proportional to the displacement from the equilibrium position, not the total stretch distance. Therefore, the spring force should be calculated using the stretch distance of 24 m, not 37 m.

Taking these factors into account, the correct calculation for the spring force would be 88 N/m * 24 m = 2112 N.

Furthermore, the maximum acceleration experienced by the bungee jumper is not simply the total acceleration due to gravity (9.8 m/s^2) added to the spring force divided by the mass. This is because the bungee cord also provides a force in the opposite direction to the acceleration due to gravity.

To calculate the maximum acceleration, we can use the equation F_net = ma, where F_net is the net force acting on the bungee jumper and a is the maximum acceleration. The net force in this case would be the spring force (2112 N) minus the weight of the bungee jumper (70 kg * 9.8 m/s^2 = 686 N).

Therefore, F_net = 2112 N - 686 N = 1426 N.

Finally, using the equation F_net = ma, we can solve for a:

1426 N = (70 kg) * a

a = 1426 N / 70 kg = 20.37 m/s^2

This is the maximum acceleration experienced by the bungee jumper, which is significantly less than the value calculated in the original attempt.

In conclusion, it is important to take into account the correct displacement and use the proper equations when solving for the spring stiffness constant and maximum acceleration in a situation involving elastic and gravitational energy conservation.