In the dangerous “sport” of bungee-jumping, a daring student jumps
from a balloon with a specially designed elastic cord attached to his
ankles. The unstretched length of the cord is 25.0 m, the student
weighs 700 N, and the balloon is 36.0 m above the surface of a river
below. Assuming that Hooke’s law describes the cord, calculate the
required force constant if the student is to stop safely 4.00 m above
the river.
4.00 m above the river surface, the bungee length is 32 m and it has stretched 7.0 m.
The stored potential energy in the cord is then
PEmax = (1/2)k*7^2 . This equals the gravitational potential energy loss, M*g*32. Kinetic energy at the top and bottom of the jump is zero.
(49/2)k = 32 M g
Solve for k.
To solve this problem, we need to apply Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Hooke's law can be represented as:
F = k * x
Where:
- F is the force exerted by the spring
- k is the force constant or spring constant
- x is the displacement from the equilibrium position
In this scenario, we can consider the bungee cord as a spring, and the displacement of the cord is equal to the difference between the unstretched length and the stretched length.
Given:
- Unstretched length of the cord (x₀) = 25.0 m
- Displacement (x) = 4.00 m
To calculate the force constant (k), we need the force exerted by the spring. Since the student jumps from a balloon, the only force acting on him is the gravitational force. We can calculate this force using the formula:
F = m * g
Where:
- m is the mass of the student
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Given:
- Weight of the student (F) = 700 N
Using the formula F = m * g, we can solve for the mass (m):
700 N = m * 9.8 m/s²
m = 700 N / 9.8 m/s² ≈ 71.43 kg
Now, we can substitute the values into Hooke's law equation and solve for the force constant (k):
F = k * x
700 N = k * 4.00 m
k = 700 N / 4.00 m ≈ 175 N/m
Therefore, the required force constant for the bungee cord is approximately 175 N/m.