James (80kg) is performing a bungee jump from a 210m high dam. He has a free fall for 160 meters until the rope starts to extend and stops 1 meter above
ground.
What is the stiffness of the bungee rope?
How much work does the bungee rope perform?
3.3
To find the stiffness of the bungee rope, we need to use Hooke's law, which states that the force applied by a spring or elastic material is directly proportional to the displacement (change in length) of the material. The formula for Hooke's law is:
F = k * x
Where:
F = Force applied by the material (stiffness of the bungee rope in this case)
k = Stiffness constant (unknown)
x = Displacement (change in length) of the material (unknown)
In this scenario, we know that the bungee rope stops extending 1 meter above the ground, which means its displacement is 160 meters (fall distance) + 1 meter (remaining distance above ground) = 161 meters.
The force applied by the bungee rope is equal to the weight of James, which is given as 80 kg. The weight can be calculated using the formula:
Weight = mass * gravity
Where:
mass = 80 kg (given)
gravity = 9.8 m/s^2 (acceleration due to gravity)
Weight = 80 kg * 9.8 m/s^2 = 784 N
Substituting the known values into Hooke's law equation:
784 N = k * 161 m
Solving for k:
k = 784 N / 161 m
k ≈ 4.865 N/m
Therefore, the stiffness of the bungee rope is approximately 4.865 N/m.
To calculate the work done by the bungee rope, we need to use the formula for work:
Work = Force * Distance
In this scenario, the force is the weight of James (784 N), and the distance is the total distance from the point where the bungee rope starts to extend (160 meters above the ground) to the point where it stops extending (1 meter above the ground), which is 160 meters.
Work = 784 N * 160 m
Work = 125,440 Joules
Therefore, the bungee rope performs approximately 125,440 Joules of work.