Falling elevator energy conservation

[kopinator]

The cable of an elevator of mass M = 1640 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 = 36.4 m above a cushioning spring whose spring constant is k = 13600 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of f = 12724 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed. (1/2)kx^2=Spring energy
K=(1/2)mv^2
U=mgh
Vf^2=Vi^2+2a(X-Xi)F=ma
mg-Ff=ma
a=2.05 m/s^2

Vf^2=Vi^2+2a(X-Xi)
Vf^2=0+2(-2.05)(-36.4)
Vf=12.22 m/s

(1/2)kx^2=(1/2)mv^2
(1/2)(13600)x^2=(1/2)(1640)(12.22^2)
x= 4.24 m (that didn't work)

(then I tried this)
(1/2)kx^2=(1/2)mv^2+mgh
x=10.20 m (still wasn't correct)

 

[ap123]

What energy changes are taking place when the elevator falls?

 

[TSny]

Don't forget that the friction force continues to act while the spring is compressing.

 

[kopinator]

Potential energy is being converted to kinetic energy. So with that, (1/2)mv^2=mgh. Would the final equation be (1/2)kx^2=mgh?

 

[ap123]

What about the friction force?

 

[kopinator]

The friction force is acting over a distance throughout the process of the elevator falling and compressing the spring. So the friction force is doing work against the falling elevator.
(1/2)kx^2=mgh-Ffd?

 

[ap123]

Ok, looking good.
Are h and d the same?
How are they related to the initial height you are given.
Hint : think what happens to the spring.

 

[kopinator]

h is the initial height and d is the initial height + the distance the spring compresses when the elevator hits it. Thus Ffd=Ff(h+x), right?

 

[ap123]

Yes, good.

 

[kopinator]

Awesome! Thank you!

 

[TSny]

h is the initial height and d is the initial height + the distance the spring compresses when the elevator hits it. Thus Ffd=Ff(h+x), right?

Does the force of gravity also act through the entire distance h+x?

 

[kopinator]

Yes it does. So will mgh be mg(h+x) instead?

 

[TSny]

Yes, I think that's right.