A 58 kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 15 m, and falls a total of 36 m.
Calculate the spring stiffness constant k of the bungee cord, assuming Hooke's law applies.
Calculate the maximum acceleration she experiences.
-mgh=-(kx^2)/2 (for first part)
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To calculate the spring stiffness constant (k) of the bungee cord, we can use Hooke's law, which states that the force applied by a spring is proportional to the displacement of the spring from its equilibrium position. The equation for Hooke's law is F = kx, where F is the force applied by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we know that the bungee cord is unstretched at a length of 15 m, and the jumper falls a total of 36 m. Since the displacement of the bungee cord is the difference between the initial and final lengths, we have x = 36 m - 15 m = 21 m.
We are also given the mass of the bungee jumper, which is 58 kg. The force applied by the bungee cord can be calculated using Newton's second law, which states that force equals mass times acceleration. In the case of the bungee jumper, the force is the weight of the jumper, which is given by the equation F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, the force applied by the bungee cord is F = (58 kg)(9.8 m/s^2) = 568.4 N.
Now we can substitute the values into Hooke's law equation to solve for the spring stiffness constant (k):
F = kx
568.4 N = k(21 m)
Rearranging the equation, we have:
k = 568.4 N / 21 m
k ≈ 27.07 N/m
So the spring stiffness constant of the bungee cord is approximately 27.07 N/m.
To calculate the maximum acceleration the jumper experiences, we can use the equation F = ma, where F is the force applied by the bungee cord and m is the mass of the jumper. We already know the force applied by the bungee cord, which is 568.4 N, and the mass of the jumper is 58 kg.
Therefore, the maximum acceleration is given by:
a = F / m = 568.4 N / 58 kg
a ≈ 9.8 m/s^2
So the maximum acceleration the jumper experiences is approximately 9.8 m/s^2, which is equal to the acceleration due to gravity.