The cable of an elevator of mass M = 1020 kg snaps when the elevator is at rest at one of the floors of a skyscraper. At this point the elevator is a distance d = 18.4 m above a cushioning spring whose spring constant is k = 6500 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of f = 6678 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed.
This is what I did but the answer was wrong!:
Wtotal = (mg- f) 18.4m
Wtotal = 1/2 kx^2
Your approach looks OK.
Check for any errors in copying the problem or any math errors.
Wtotal = [(1020)(9.8)-6678] 18.4
= 61051.2J
61051.2J = 1/2 (6500)x^2
x = 4.33m
STill wrong :( Help
The displacement of the safety spring has to be considered in the total potential energy of the elevator car.
Um are you saying
Wtotal = [(1020)(9.8)-6678] (18.4+x) ?..
In that case, I did the quadratic and got x=17.33m, but still wrong answer :( argg
For the roots of the quadratic I got
+4.875
-3.854
To solve this problem, we can use the principle of conservation of mechanical energy.
First, let's consider the initial state of the elevator when the cable snaps. At this point, the elevator is at rest, so its initial kinetic energy is zero. The only form of energy present is gravitational potential energy given by mgh, where m is the mass of the elevator, g is the acceleration due to gravity, and h is the height above the cushioning spring.
So, the initial mechanical energy of the elevator is given by the gravitational potential energy:
E_initial = mgh = (1020 kg)(9.8 m/s^2)(18.4 m)
Next, let's consider the final state of the elevator when it comes to rest after compressing the cushioning spring. In this state, the elevator has zero kinetic energy, and the only form of energy present is the potential energy stored in the compressed spring.
The potential energy stored in the spring is given by:
E_spring = 1/2 kx^2, where k is the spring constant, and x is the maximum distance by which the cushioning spring will be compressed.
Now, using the conservation of mechanical energy, we can equate the initial and final energies:
E_initial = E_spring
mgh = 1/2 kx^2
Substituting the given values:
(1020 kg)(9.8 m/s^2)(18.4 m) = 1/2 (6500 N/m) x^2
Simplifying the equation, we can solve for x:
(183696 N) = (3250 N/m) x^2
x^2 = (183696 N) / (3250 N/m)
x^2 = 56.45 m^2
x = sqrt(56.45) m
x ≈ 7.52 m
Therefore, the maximum distance by which the cushioning spring will be compressed is approximately 7.52 meters.