Solving for Spring Constant and Distance in Falling Elevators Problem

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suffian
edit::
nm, i figured it out. i forgot to account for the weight of elevator in the "If static friction.. " part. thanks anyway. I'm sorry i can't delete the post.
::edit

I hit a brick wall over yet another problem concerning energy methods, basically a mismatch between my answer and the book's. I would appreciate whoever has the time to look this over and guess where I might be going wrong.

Anyway, here's the problem verbatim:

Redesign the elevator safety system of Example 7-11 [A falling elevator stopped by a powerful spring] so that the elevator does not bounce but stays at rest the first time its speed becomes zero. The mass of the elevator is 2000 kg and its speed when it first touches the spring is 25 m/s. There is a kinetic friction force of 17,000 N and the maximum static friction on the elevator is also 17,000 N. The mass of the spring can be neglected. a) What spring constant is required, and what distance is the spring compressed when the elevator is stopped? Do you think the design is practical? Explain. b) What is the maximum magnitude of the acceleration of the elevator?

Book Ans: a) 919 N/m, 39.8 m b) 17.0 m/s^2

Begin with work-energy formula:
[tex] \begin{align*}
\Delta K =& \, W_\text{gravity} + W_\text{friction} + W_\text{spring} \\
0 - \frac{1}{2}mv^2 =&\, mgX - fX - \frac{1}{2}kX^2
\end{align*} [/tex]

Quadratic formula to find compression distance X (neglect neg ans):
[tex]
X = \frac{1}{k} (mg - f + \sqrt{ (mg-f)^2 +kmv^2 })
[/tex]

If static friction will hold elevator at X, then it follows:
[tex] F = kX \leq f \text{ or } X \leq \frac{f}{k} [/tex]
[tex]
X = \frac{1}{k} (mg - f + \sqrt{ (mg-f)^2 +kmv^2 }) \leq \frac{f}{k} [/tex]

Manipulate to find assertion about spring constant:
[tex] k \leq \frac{f}{mv^2} (3f - 2mg) [/tex]
[tex]
k \leq \frac{[17000 \text{ N}]}{[2000 \text{ kg}][25 \text{ m/s}]^2} (3[17000 \text{ N}] - 2[2000 \text{ kg}][9.80 \text{ m/ss}])
[/tex]
[tex] k \leq 160 \text{ N/m} [/tex]

So, answer is not the same.

edit::
nm, i figured it out. i forgot to account for the weight of elevator in the "If static friction.. " part. thanks anyway. I'm sorry i can't delete the post.
::edit
 
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  • #2


I am glad to hear that you were able to figure out the solution to the problem you were having. It can be frustrating when our answers do not match up with the expected results, but it is always a good learning opportunity to go back and check our calculations.

In this case, it seems like you forgot to take into account the weight of the elevator when determining the maximum distance that the elevator can be held by static friction. It is important to always consider all forces acting on an object when using energy methods.

I hope this helps and good luck with your future problem-solving. Keep up the good work as a scientist!
 
  • #3


Great job on solving the problem! It's always a good feeling when we figure out where we went wrong and are able to correct it. Don't worry about not being able to delete the post, it's a learning process and others may benefit from seeing how you approached and solved the problem. Keep up the good work!
 

1. How do you determine the spring constant in a falling elevator problem?

The spring constant in a falling elevator problem can be determined by using Hooke's law, which states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. This means that by measuring the displacement of the spring and the corresponding force, the spring constant can be calculated.

2. What is the equation to solve for the distance in a falling elevator problem?

The equation to solve for the distance in a falling elevator problem is d = 1/2 * g * t^2, where d is the distance, g is the acceleration due to gravity, and t is the time.

3. Can the spring constant and distance in a falling elevator problem be solved without knowing the mass of the elevator?

Yes, the spring constant and distance can be solved without knowing the mass of the elevator. This is because the mass of the elevator does not affect the acceleration due to gravity, which is the key factor in solving for these variables.

4. What is the role of the spring in a falling elevator problem?

The spring in a falling elevator problem serves as a means to measure the force and displacement of the elevator. It acts as a sensor that can provide data to solve for the spring constant and distance.

5. How does air resistance affect the calculation of the spring constant and distance in a falling elevator problem?

Air resistance does not significantly affect the calculation of the spring constant and distance in a falling elevator problem. This is because the effect of air resistance is minimal in short periods of time, which is the case in most falling elevator problems.

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