A 300 lb bungee jumper wants to jump from the top of a 230 ft high bridge and be able to just touch the water below (Part A figure) . Given that the bungee cord has a spring constant of 11.1 lb/ft, what should the bungee cord's relaxed length, l, be for this jump?
cord potential energy at maximum allowed extension = potential energh loss of jumper, at the bottom
(1/2)k(230-L)^2 = M*g*230
Solve for L.
k is the spring constant.
Mass must be in slugs, or use M*g = 300 lb.
To calculate the relaxed length of the bungee cord, we need to use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
First, let's convert the weight of the bungee jumper from pounds to Newtons:
Weight = mass * acceleration due to gravity
Weight = (300 lb) * (4.448 N/lb) = 1334.4 N
Next, we need to find the displacement of the bungee jumper from the relaxed length. From the information given, the bungee jumper wants to touch the water, so the displacement would be the height of the bridge minus the height of the water:
Displacement = 230 ft - h_water
Now, we can calculate the force exerted by the bungee cord using Hooke's law:
Force = -k * displacement
where k is the spring constant of the bungee cord and displacement is the change in length from the relaxed length.
Since the displacement is negative (the bungee cord will be stretched), we need to include a negative sign in the equation.
Now, let's plug in the values we have:
1334.4 N = - (11.1 lb/ft) * (230 ft - h_water)
Simplifying the equation:
1334.4 N = - (11.1 lb/ft) * (230 ft - h_water)
1334.4 N = - (11.1 lb/ft) * 230 ft + (11.1 lb/ft) * h_water
1334.4 N = -2553 lb + (11.1 lb/ft) * h_water
Next, we can solve for h_water:
h_water = (1334.4 N + 2553 lb) / (11.1 lb/ft)
Now, we can substitute the value of h_water back into the displacement equation to find the relaxed length:
Displacement = 230 ft - h_water
l = 230 ft - h_water
Substituting the value of h_water, we can calculate l.
Calculation:
h_water = (1334.4 N + 2553 lb) / (11.1 lb/ft)
≈ 279.82 ft
l = 230 ft - h_water
= 230 ft - 279.82 ft
≈ -49.82 ft
Since the length cannot be negative, we take the absolute value:
l ≈ 49.82 ft
Therefore, the relaxed length of the bungee cord should be approximately 49.82 ft.
To determine the bungee cord's relaxed length (l) for a 300 lb bungee jumper jumping from a 230 ft high bridge, we need to consider the equilibrium position where the jumper just touches the water.
Let's begin by calculating the force exerted by the bungee cord when the jumper is at the equilibrium position. The force exerted by the bungee cord is given by Hooke's Law:
F = k * x
Where:
F is the force
k is the spring constant
x is the displacement from the relaxed length
In this case, the force (F) exerted by the bungee cord is equal to the weight of the jumper, which is 300 lb. The spring constant (k) is given as 11.1 lb/ft.
At the equilibrium position, the displacement (x) will be equal to l (the relaxed length) plus the distance the jumper has fallen, which in this case is 230 ft. So, x = l + 230.
Substituting the given values into the equation, we have:
300 lb = 11.1 lb/ft * (l + 230 ft)
Now, we can solve for l (the relaxed length):
300 lb = 11.1 lb/ft * l + 11.1 lb/ft * 230 ft
300 lb - 11.1 lb/ft * 230 ft = 11.1 lb/ft * l
Subtracting 11.1 lb/ft * 230 ft from both sides:
300 lb - 11.1 lb/ft * 230 ft = 11.1 lb/ft * l - 11.1 lb/ft * 230 ft
300 lb - 11.1 lb/ft * 230 ft = 11.1 lb/ft * (l - 230 ft)
Next, let's isolate l by dividing both sides of the equation by 11.1 lb/ft:
(300 lb - 11.1 lb/ft * 230 ft) / (11.1 lb/ft) = l - 230 ft
Finally, adding 230 ft to both sides, we get the value of l (the relaxed length):
(300 lb - 11.1 lb/ft * 230 ft) / (11.1 lb/ft) + 230 ft = l
Evaluating the equation will give us the required relaxed length for the bungee cord in order for the jumper to just touch the water.