A bungee jumper with mass 63.5 kg jumps from a high bridge. After reaching his lowest point, he oscillates up and down, hitting a low point eight more times in 40.0 s. He finally comes to rest 28.5 m below the level of the bridge. Calculate the spring stiffness constant AND the unstretched length of the bungee cord.
To calculate the spring stiffness constant and the unstretched length of the bungee cord, we can use the equation for the period of a simple harmonic oscillator:
T = 2π√(m/k)
Where:
T = Period of oscillation
m = Mass of the bungee jumper
k = Spring stiffness constant
First, let's calculate the period of oscillation.
Given:
Number of oscillations = 8
Time for those oscillations = 40.0 s
We know that one complete oscillation consists of two low points. Therefore, the total number of low points is twice the number of oscillations.
Total number of low points = 2 x 8 = 16
The time for these 16 low points is twice the given time because each oscillation has two low points.
Time for 16 low points = 2 x 40.0 s = 80.0 s
Now, divide the total time by the number of oscillations to get the period of oscillation:
T = 80.0 s / 16 = 5.0 s
Next, let's calculate the acceleration due to gravity.
Given:
Height of the lowest point = 28.5 m
At the lowest point of the oscillation, the bungee jumper is momentarily at rest. This means that the gravitational force pulling him downward is equal to the spring force pulling him upward.
Therefore, we can write the equation:
mg = kΔL
Where:
m = Mass of the bungee jumper
g = Acceleration due to gravity (9.8 m/s^2)
k = Spring stiffness constant
ΔL = Change in length of the bungee cord (stretched length - unstretched length)
Solving for ΔL, we get:
ΔL = mg/k
Now, let's substitute the known values into the equation:
ΔL = (63.5 kg) * (9.8 m/s^2) / k
The change in length of the bungee cord, ΔL, is equal to twice the distance from the lowest point to the level of the bridge:
ΔL = 2 * 28.5 m = 57.0 m
Setting the two equations for ΔL equal to each other:
mg/k = 57.0 m
Now, solve for k:
k = mg / 57.0 m
Substituting the known values:
k = (63.5 kg) * (9.8 m/s^2) / 57.0 m
Finally, we have k, the spring stiffness constant. To find the unstretched length of the bungee cord (L0), we need to solve for ΔL:
ΔL = mg/k
Now, substitute the known values:
ΔL = (63.5 kg) * (9.8 m/s^2) / k
Finally, subtract ΔL from the lowest point height to find the unstretched length:
L0 = 28.5 m - ΔL