A 56 kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 11 m, and falls a total of 38 m. Calculate the spring stiffness constant k of the bungee cord, assuming Hooke's law applies. Calculate the maximum acceleration she experiences.
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A)
GPE lost is mgh = 55 x 9.8 x 31 = 16709 J
this turns into strain energy in the bungee cord (assuming no friction, sound, etc)
strain energy = 1/2 k x^2
x = 31 - 13 = 18 m
so 16709 = 1/2 * k * (18)^2 = 162 k
so k = 16709 = 103 N m^-1 = 100 to 2 sf (well, 103 is 3 sf so what can I do?)
B)
acceleration = F / m
F = k x = 103 * 18 = 1854 N
so acceleration = 1854 / 55 = 33.7 m s^-2 = 33 m s^-2 to 2 sf
final acceleration minus 9.8m/s^2 is the maximum acceleration because gravity is always there to pull you down
To calculate the spring stiffness constant (k) of the bungee cord, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to its displacement. The formula for Hooke's law is:
F = k * x
where F is the force exerted by the spring, k is the spring stiffness constant, and x is the displacement from the spring's equilibrium position.
In this case, the force exerted by the bungee cord is equal to the weight of the bungee jumper. We can calculate the force as:
F = m * g
where m is the mass of the bungee jumper and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F = 56 kg * 9.8 m/s^2
F = 549.6 N
Next, we need to find the displacement of the bungee cord from its unstretched length. The total length of the fall is 38 m, so the displacement is:
x = total length - unstretched length
x = 38 m - 11 m
x = 27 m
Now, we can substitute the values into Hooke's law and solve for the spring stiffness constant, k:
549.6 N = k * 27 m
Dividing both sides of the equation by 27 m:
k = 549.6 N / 27 m
k ā 20.36 N/m
Therefore, the spring stiffness constant (k) of the bungee cord is approximately 20.36 N/m.
To calculate the maximum acceleration the bungee jumper experiences, we can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass. The formula is:
a = F / m
In this case, the net force is given by the force exerted by the bungee cord minus the force of gravity:
Net force = F - mg
Plugging in the values:
Net force = 549.6 N - (56 kg * 9.8 m/s^2)
Net force = 549.6 N - 548.8 N
Net force = 0.8 N
Now we can calculate the maximum acceleration:
a = 0.8 N / 56 kg
a ā 0.0143 m/s^2
Therefore, the maximum acceleration the bungee jumper experiences is approximately 0.0143 m/s^2.
To find the spring stiffness constant (k) of the bungee cord, we'll need to apply Hooke's law. Hooke's law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
We'll start by calculating the net force acting on the bungee jumper at the maximum displacement (38 m). At this point, the gravitational force pulling the jumper downward is balanced by the spring force pulling her upward.
First, let's find the weight (W) of the bungee jumper using the formula:
W = mass * gravitational acceleration
W = 56 kg * 9.8 m/s^2
W = 548.8 N
At the maximum displacement, the spring force (Fs) will be equal in magnitude to the weight:
Fs = W = 548.8 N
Next, we can calculate the spring stiffness constant (k) using the formula:
k = Fs / displacement
k = 548.8 N / 11 m
k ā 49.88 N/m
Therefore, the spring stiffness constant (k) of the bungee cord is approximately 49.88 N/m.
Now, to find the maximum acceleration, we can use the equation:
max acceleration = (Fs - W) / mass
max acceleration = (548.8 N - 548.8 N) / 56 kg
max acceleration = 0 m/s^2
The maximum acceleration experienced by the bungee jumper is 0 m/sĀ², indicating that she reaches a maximum deceleration but no acceleration.