The world’s tallest building is the Burj Khalifa which stands at 828 above the ground. An eccentric billionaire CEO has an office on the top floor. He insists on having a personal elevator installed that consists only of a giant spring that spans from the basement to his office when it is uncompressed. If his mass is 120kg, what spring constant in N/m is required so that he momentarily comes to rest on the ground floor?
To find the spring constant required for the eccentric billionaire CEO to momentarily come to rest on the ground floor, we need to consider the gravitational force acting on him and the elastic force exerted by the spring.
The gravitational force acting on the CEO is given by the equation:
F_gravity = m * g
Where:
F_gravity is the force due to gravity
m is the mass of the CEO (120 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Next, we need to find the compression of the spring. Since the CEO momentarily comes to rest, we can assume that the total force exerted by the spring at this point is equal in magnitude but opposite in direction to the gravitational force. Therefore, we can equate the two forces:
F_spring = F_gravity
The force exerted by a spring is given by Hooke's Law:
F_spring = -k * x
Where:
F_spring is the force exerted by the spring
k is the spring constant (to be determined)
x is the compression of the spring (to be determined)
Since the CEO momentarily comes to rest, the acceleration is zero. Therefore, the sum of the forces must be zero:
F_spring + F_gravity = 0
Plugging in the equations and values we have:
-k * x + m * g = 0
Simplifying the equation:
k * x = m * g
Finally, we can solve for the spring constant:
k = (m * g) / x
Given that the height of the Burj Khalifa is 828 meters, and the uncompressed length of the spring is equal to this height, the compression of the spring would be:
x = 828 m
Plugging in the values, we get:
k = (120 kg * 9.8 m/s^2) / 828 m
Simplifying:
k ≈ 1.430 N/m
Therefore, a spring constant of approximately 1.430 N/m is required for the eccentric billionaire CEO to momentarily come to rest on the ground floor with a personal elevator consisting of a giant spring.