A bungee jumper with mass 61.3 kg jumps from a high bridge. After reaching his lowest point, he oscillates up and down, hitting a low point eight more times in 31.8 s. He finally comes to rest 25.3 m below the level of the bridge. Calculate the spring constant of the bungee cord. What is the unstretched length of the bungee cord?
i do not know
To calculate the spring constant of the bungee cord, we can first find the period of oscillation, and then use it to determine the spring constant.
Step 1: Find the period of oscillation.
The time it takes for the bungee jumper to hit a low point and return to that same position is called the period (T) of the oscillation. In this case, we are given that the bungee jumper hits a low point eight more times in 31.8 s. Since there is one complete oscillation between each low point, we can calculate the period using the following formula:
T = Total time / Number of oscillations
T = 31.8 s / 8 = 3.975 s
Step 2: Calculate the spring constant.
The period of oscillation is related to the spring constant (k) and the mass of the bungee jumper (m) through the following formula:
T = 2π√(m/k)
Rearranging the formula, we can solve for the spring constant (k):
k = (4π²m) / T² = (4π²(61.3 kg)) / (3.975 s)²
Using the value of π as approximately 3.14159, we can now calculate the spring constant.
k ≈ (4 * 3.14159 * 61.3 kg) / (3.975 s)²
k ≈ 242.025 N/m
Therefore, the spring constant of the bungee cord is approximately 242.025 N/m.
Step 3: Calculate the unstretched length.
To calculate the unstretched length of the bungee cord, we need to determine the maximum extension of the cord when the bungee jumper comes to rest 25.3 m below the level of the bridge.
The maximum extension of the cord occurs when the gravitational potential energy is equal to the energy stored in the spring. Using the formula for gravitational potential energy:
m * g * h = (1/2) * k * x²
Where:
m = mass of the bungee jumper (61.3 kg)
g = acceleration due to gravity (9.8 m/s²)
h = height the bungee jumper falls (25.3 m)
k = spring constant (242.025 N/m)
x = extension of the bungee cord (unstretched length)
Rearranging the formula, we can solve for x:
x = √((2 * m * g * h) / k)
x = √((2 * 61.3 kg * 9.8 m/s² * 25.3 m) / 242.025 N/m)
x ≈ 3.028 m
Therefore, the unstretched length of the bungee cord is approximately 3.028 m.