Falling elevator energy conservation

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  • #1
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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)
 

Answers and Replies

  • #2
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What energy changes are taking place when the elevator falls?
 
  • #3
TSny
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Don't forget that the friction force continues to act while the spring is compressing.
 
  • #4
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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?
 
  • #5
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What about the friction force?
 
  • #6
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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?
 
  • #7
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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.
 
  • #8
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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?
 
  • #9
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Yes, good.
 
  • #10
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Awesome! Thank you!
 
  • #11
TSny
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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?
 
  • #12
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Yes it does. So will mgh be mg(h+x) instead?
 
  • #13
TSny
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Yes, I think that's right.
 

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