In the dangerous "sport" of bungee-jumping, a daring student jumps from a balloon with a specially designed elastic cord attached to his waist, as shown in Figure P5.77. The unstretched length of the cord is 25.0 m, the student weighs 675 N, and the balloon is 36.0 m above the surface of a river below. Calculate the required force constant (k) of the cord if the student is to stop safely 4.00 m above the river.
Sorry it is to HARD!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
To calculate the required force constant of the cord (k), we need to consider the potential energy and the force exerted by the cord on the student.
1. First, let's calculate the initial potential energy of the student before he jumps. The potential energy is given by the formula:
PE_initial = m * g * h
where m is the mass of the student, g is the acceleration due to gravity, and h is the height of the balloon above the river.
In this case, the weight of the student (675 N) is equal to the force of gravity (m * g), so we can use it directly in the formula. Thus, the initial potential energy is:
PE_initial = 675 N * 36.0 m
2. Next, let's calculate the final potential energy of the student when he stops safely 4.00 m above the river. The final potential energy is given by the formula:
PE_final = m * g * h'
where h' is the final height above the river.
In this case, the final potential energy is:
PE_final = 675 N * 4.00 m
3. The difference between the initial and final potential energy represents the work done by the cord in stopping the student's fall safely. We can calculate the work using the formula:
Work = PE_initial - PE_final
4. The elastic potential energy stored in the cord is given by the formula:
PE_elastic = (1/2) * k * x^2
where k is the force constant of the cord and x is the distance the cord is stretched.
In this case, the change in length of the cord is:
x = (25.0 m + 4.00 m) - 25.0 m = 4.00 m
Substituting the given values into the formula, we get:
PE_elastic = (1/2) * k * (4.00 m)^2
5. The work done by the cord is equal to the elastic potential energy stored:
Work = PE_elastic
Substituting the expressions for Work and PE_elastic, we have:
PE_initial - PE_final = (1/2) * k * (4.00 m)^2
6. Now, solve the equation for k:
k = 2 * (PE_initial - PE_final) / (4.00 m)^2
Substituting the given values into the formula, we have:
k = 2 * (675 N * 36.0 m - 675 N * 4.00 m) / (4.00 m)^2
Simplifying the equation gives:
k = 2 * 675 N * (36.0 m - 4.00 m) / (4.00 m)^2
Finally, calculate the result to find the required force constant (k).