A bungee jumper with mass 69.7 kg jumps from a high bridge. After reaching his lowest point, he oscillates up and down, hitting a low point eight more times in 39.0 s. He finally comes to rest 25.0 m below the level of the bridge. Calculate the spring constant of the bungee cord.
To calculate the spring constant of the bungee cord, we need to use the equation that relates the period of oscillation to the spring constant. The period is the time it takes for the jumper to complete one full oscillation.
First, let's calculate the period of oscillation. Since the bungee jumper hits a low point eight more times in 39.0 s after reaching the lowest point, we can calculate the average time it takes for one oscillation.
Average time per oscillation = Total time / Number of oscillations
Average time per oscillation = 39.0 s / 9 oscillations
Average time per oscillation = 4.333 s
Next, we use the formula that relates the spring constant (k) to the period (T):
T = 2π * √(m / k)
Where:
T is the period of oscillation
π is a mathematical constant, approximately equal to 3.14159
m is the mass of the bungee jumper
k is the spring constant of the bungee cord
Rearranging the formula, we get:
k = (4π² * m) / T²
Substituting the known values:
m = 69.7 kg (mass of the bungee jumper)
T = 4.333 s (average time per oscillation)
k = (4 * 3.14159² * 69.7) / (4.333²)
Now we can calculate:
k = (4 * 3.14159² * 69.7) / (4.333²)
k ≈ 487 N/m
Therefore, the spring constant of the bungee cord is approximately 487 N/m.