Hooke's Law & Energy conservation

In summary, an engineer is designing a spring to be placed at the bottom of an elevator shaft and must calculate the spring constant in order to limit the passengers' acceleration to no more than 5.0 g when the elevator cable breaks. Using Hooke's law and energy conservation, the value of k is found to be 12mg/h.
  • #1
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Homework Statement



An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable should happen to break when the elevator is at a height h above the top of the spring, calculate the value of the spring constant k so that the passengers undergo an acceleration of no more than 5.0 g when brought to a rest. Let M be the total mass of the elevator and passengers.

Homework Equations



F=-kx, Hooke's law
Energy conservation: spring energy, gravitational energy

The Attempt at a Solution



The maximum acceleration will occur at the maximum compression of the spring, because a is proportional to x. Since kx=ma=5mg, k=5mg/x.

I then conserved energy to get 1/2kx^2 - mgx - mgh = 0.

I solved both equations to get k = 15mg/(2h). But my solution key says 12mg/h. What's wrong?

EDIT: Never mind, solved it. My force equation was off.
 
Last edited:
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  • #2
Well done :)
Perhaps you could show others where you went wrong with the force equation?
 

1. What is Hooke's Law?

Hooke's Law is a principle in physics that states that the force needed to extend or compress a spring is directly proportional to the distance that the spring is stretched or compressed.

2. What is the equation for Hooke's Law?

The equation for Hooke's Law is F = -kx, where F is the force applied, k is the spring constant, and x is the displacement of the spring from its equilibrium position.

3. How is energy conserved in Hooke's Law?

In Hooke's Law, energy is conserved because the work done by the force applied to the spring is equal to the potential energy stored in the spring. This means that any energy put into the system by stretching or compressing the spring is stored as potential energy and can be released when the spring returns to its equilibrium position.

4. How does Hooke's Law relate to simple harmonic motion?

Hooke's Law is the basis for simple harmonic motion, which is the oscillatory motion of a system where the force is directly proportional to the displacement from the equilibrium position. This means that Hooke's Law governs the motion of systems such as mass-spring systems and pendulums.

5. What are some real-life applications of Hooke's Law?

Hooke's Law has many practical applications, including in the design of springs for mechanical devices, such as shock absorbers, car suspensions, and mattresses. It is also used in measuring devices, such as spring scales and force gauges, and in medical devices, such as prosthetic limbs and braces.

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