Conservation of Energy With Loss Due to Friction

In summary, the problem involves an elevator of mass 2690 kg at rest, 75.0 m above a cushioning spring with spring constant 9700 N/m. The elevator snaps and a safety device applies a constant frictional force of 9358 N. The maximum distance the spring will be compressed can be found by setting the work done by the spring equal to the initial potential energy minus the work done by friction. This gives a quadratic equation, which can be solved to find the maximum distance. It is important to include the work done by friction throughout the motion, not just over the distance x.
  • #1
seichan
32
0
[SOLVED] Conservation of Energy With Loss Due to Friction

Homework Statement



The cable of an elevator of mass m= 2690 kg snaps when the elevator is a rest at one of the floors of a skyscraper. At this point the elevator is a distance d= 75.0 m above a cushioning spring whose spring constant is k= 9700 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of f= 9358 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed.

Homework Equations


U(x)=mgh
w(x)=1/2kx^2


The Attempt at a Solution



Alright, I thought this problew was pretty straight forward. I first calculated the potential energy for the elevator when it is at rest, in which I found to be U(x)=mgd. This would be equal to the Kinetic friction at the bottom, however, there is energy loss to friction. So, the energy when the elevator hits the spring is U(x)-Force of friction, or mgd-f. We can set this eqaul to the amount of work done by the spring and solve for x. In other words: sqrt((2(mgd-f))/k)=x. The displacement is negative, so we want the negative value of this square root. I have put this answer into my web assignment a few times and cannot figure out where I am going wrong (I've tried using a positive value for the displacement as well). If you could tell me where my reasoning is wrong, that would be great.
 
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  • #2
When calculating the gravitational PE, don't forget to include the distance that the spring compresses.
 
  • #3
Doc Al said:
When calculating the gravitational PE, don't forget to include the distance that the spring compresses.

Thank you very much- but how do I isolate x in that case?

1/2kx^2=mg(d+x)-f
=mgd+mgx-f
1/2kx^2-mgx=mgd-f

Also, can't I look at the top of the spring as the x-axis? That was the asumption I was going with.
 
  • #4
seichan said:
Thank you very much- but how do I isolate x in that case?

1/2kx^2=mg(d+x)-f
=mgd+mgx-f
1/2kx^2-mgx=mgd-f
That "f" should be the work done by friction, not just the friction force.

You'll end up with a quadratic equation. Solve it!

Also, can't I look at the top of the spring as the x-axis? That was the asumption I was going with.
Not sure what you mean. You can certainly measure distances from the top of the spring, but in that case your final position will have a gravitational PE term. (If that's what you mean.)
 
  • #5
To clarify, it should be:
1/2kx^2=mg(d+x)-fx
1/2kx^2=mgd+(mg-f)x
1/2kx^2-(mg-f)x-mgd=0
Right? Thank you for your help =)
 
  • #6
The friction force acts throughout the motion, not just over the distance x. Redo the work done against friction.
 
  • #7
Thank you so much =) It's correct
 

1. What is the conservation of energy?

The conservation of energy is a fundamental principle in physics that states that energy cannot be created or destroyed, but can be transferred from one form to another.

2. How does friction impact the conservation of energy?

Friction is a force that opposes motion between two surfaces in contact, and it can cause energy to be converted into heat and lost from a system. This reduces the overall amount of energy available for use and can impact the conservation of energy.

3. Can energy be conserved with friction present?

Yes, energy can still be conserved even with friction present. This is because the total amount of energy in a closed system remains constant, even if some energy is lost due to friction.

4. What are some examples of energy loss due to friction?

Examples of energy loss due to friction include the heat generated when rubbing your hands together, the loss of energy in a car's engine due to friction between moving parts, and the loss of energy in a pendulum due to friction at its pivot point.

5. How can we reduce energy loss due to friction?

There are several ways to reduce energy loss due to friction, including using lubricants to reduce contact between surfaces, using smoother surfaces, and minimizing the amount of friction by reducing the weight or speed of objects in motion.

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