As part of a fundraiser, you want the new dean to bungee jump from a crane. The jump will be made from 44 m above a 2.5 m deep pool of Jello. A 30 m long bungee cord would be attached to the dean's ankle. You must convince the dean that your plan is safe for a person of his mass, 70 kg. As the bungee cord stretches, it will exert a force with the same properties as the force exerted by a spring. Your plan has the dean stepping off a platform and being in free fall for the 30 m before the cord begins to stretch. You must determine the elastic constant of the bungee cord so that it stretches only 12 m, which will keep the dean's head just out of the Jello.
(a) What is the algebraic expression for the spring constant of the bungee cord in terms of the unstretched length of the cord (L), the height of the jump above the pool (H), the amount the cord stretches (S), the mass of the dean (M), and the gravitational field strength (g)? [Note: Don't enter an equation like "x=blah". Just enter the "blah" part. All letters are capital except for "g".]
(b) What is the numerical value of the spring constant (make sure to include units and put a space between the number and the units)?
(a) The algebraic expression for the spring constant of the bungee cord can be given as:
k = (M * g) / S
where:
k is the spring constant,
M is the mass of the dean,
g is the gravitational field strength, and
S is the amount the cord stretches.
(b) To determine the numerical value of the spring constant, we need to use the given values:
- Height of the jump above the pool (H) = 44 m
- Depth of the pool (Jello) = 2.5 m
- Length of the bungee cord (L) = 30 m
- Amount the cord stretches (S) = 12 m
- Mass of the dean (M) = 70 kg
- Gravitational field strength (g) = 9.8 m/s^2
The unstretched length of the cord (L) is not used in the calculation of the spring constant.
Using the formula, we can calculate the spring constant (k):
k = (M * g) / S
k = (70 kg * 9.8 m/s^2) / 12 m
k ≈ 57.83 N/m
Therefore, the numerical value of the spring constant is approximately 57.83 N/m.
(a) The spring constant of the bungee cord can be expressed as:
k = (M * g) / ((L + S) - H)
Where:
k is the spring constant
M is the mass of the dean (70 kg)
g is the gravitational field strength (9.8 m/s^2)
L is the unstretched length of the cord (30 m)
S is the amount the cord stretches (12 m)
H is the height of the jump above the pool (44 m)
(b) To find the numerical value of the spring constant, we can substitute the given values into the equation:
k = (70 kg * 9.8 m/s^2) / ((30 m + 12 m) - 44 m)
k = (686 N) / (42 m - 44 m)
k = 686 N / -2 m
k = -343 N/m
The spring constant of the bungee cord is approximately 343 N/m (Newtons per meter).