A 66 kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 14 m, and falls a total of 34m.
1. Calculate the spring stiffness constant of the bungee cord, assuming Hooke's law applies.
2. Calculate the maximum acceleration she experiences.
To calculate the spring stiffness constant of the bungee cord, we will use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
1. Spring stiffness constant (k) is given by the formula k = F / x, where F is the force exerted by the spring and x is the displacement.
Since the jumper falls a total of 34m, the displacement (x) will be 34m - 14m = 20m (subtracting the unstretched length of the bungee cord).
The force exerted by the bungee cord can be calculated using the weight of the jumper. The weight (W) is given by the formula W = mass * gravity, where mass is 66 kg and gravity is 9.8 m/s^2.
W = 66 kg * 9.8 m/s^2 = 646.8 N
Therefore, the spring stiffness constant (k) = F / x = 646.8 N / 20m = 32.34 N/m
2. To calculate the maximum acceleration, we will use the equation of motion for an object experiencing a constant force. The equation is given by F = m * a, where F is the force, m is the mass, and a is the acceleration.
The force (F) exerted by the bungee cord can be calculated using Hooke's law: F = k * x, where k is the spring stiffness constant and x is the displacement.
At the maximum acceleration, the force exerted by the bungee cord will be equal to the weight of the jumper (since there are no other forces acting on the jumper).
So, F = W = 646.8 N
Using F = k * x, we can rearrange the equation to find the maximum acceleration (a): a = F / m
a = 646.8 N / 66 kg ≈ 9.80 m/s^2
Therefore, the maximum acceleration the jumper experiences is approximately 9.80 m/s^2.
To calculate the spring stiffness constant of the bungee cord, we can use Hooke's law. Hooke's law states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its equilibrium position.
1. Calculate the spring stiffness constant (k):
The force F acting on the bungee cord at any point is given by:
F = k * x
Where:
F is the force applied on the cord (weight = mass * acceleration due to gravity),
k is the spring stiffness constant,
x is the displacement of the cord from its unstretched length.
Since the force applied on the cord is equal to the weight, we can calculate it using the mass and acceleration due to gravity formula:
Weight = mass * acceleration due to gravity = 66 kg * 9.8 m/s^2
Now, let's calculate the displacement (x) of the bungee cord.
The total length of the bungee cord is the sum of the unstretched length and the displacement:
Total length = unstretched length + displacement
Given that the total length is 34 m and the unstretched length is 14 m, we can calculate the displacement:
Displacement (x) = Total length - Unstretched length = 34 m - 14 m
Now, we have all the necessary values to calculate the spring stiffness constant:
F = k * x
k = F / x
Plugging in the values:
k = (weight) / (displacement)
Calculate the weight:
Weight = mass * acceleration due to gravity = 66 kg * 9.8 m/s^2
Calculate the displacement:
Displacement (x) = 34 m - 14 m
Finally, calculate the spring stiffness constant:
k = Weight / Displacement
2. To calculate the maximum acceleration experienced by the bungee jumper, we can use the equation of motion for a spring:
a = (F - mg) / m
Where:
a is the acceleration,
F is the force applied on the cord,
m is the mass of the bungee jumper,
g is the acceleration due to gravity.
We already have the force applied on the cord (weight) and the mass of the bungee jumper. The maximum acceleration occurs when the bungee cord is fully stretched, i.e., at the maximum displacement. So we can use the maximum displacement value obtained earlier.
Plug in the values in the equation:
a = (F - mg) / m
Calculate the weight:
Weight = mass * acceleration due to gravity = 66 kg * 9.8 m/s^2
Use the value of the weight and mass in the equation to calculate the maximum acceleration.