Hooke's Law vs. Conservation of Energy

In summary, in question 1, if the equilibrium position is unknown, Hooke's law can be used to find the force on the system. In question 2, if the equilibrium position is unknown, using energy to calculate the force would give the wrong answer.
  • #1
keroberous
15
1
Homework Statement
I can never quite remember when solving problems with elastic materials whether I should be using Hooke's Law or conservation of energy. Below are two sample problems.
Relevant Equations
Hooke's Law: ##F=kx##
Elastic Potential Energy: ##E_e=\frac{1}{2}kx^2##
Gravitational Potential Energy: ##E_g=mgh##
Here are the two questions I want to compare:

1. A student of mass 62 kg stands on an upholstered chair containing springs, each of force constant 2.4 × 103 N/m. If the student is supported equally by six springs, what is the compression of each spring?

2. A 0.20-kg ball attached to a vertical spring of force constant 28 N/m is released from rest from the unstretched equilibrium position of the spring. Determine how far the ball falls, under negligible air resistance, before being brought to a momentary stop by the spring.

I can get the right answer for both questions, by using Hooke's law for number 1 and conservation of energy for question 2. My question is if I didn't know what the answers were, how would I choose the method? By using the incorrect method I'm off by a factor of two in both cases (bigger in 1 and smaller in 2).
 
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  • #2
keroberous said:
Here are the two questions I want to compare:

1. A student of mass 62 kg stands on an upholstered chair containing springs, each of force constant 2.4 × 103 N/m. If the student is supported equally by six springs, what is the compression of each spring?

2. A 0.20-kg ball attached to a vertical spring of force constant 28 N/m is released from rest from the unstretched equilibrium position of the spring. Determine how far the ball falls, under negligible air resistance, before being brought to a momentary stop by the spring.

I can get the right answer for both questions, by using Hooke's law for number 1 and conservation of energy for question 2. My question is if I didn't know what the answers were, how would I choose the method? By using the incorrect method I'm off by a factor of two in both cases (bigger in 1 and smaller in 2).
In question 1, you are asked to determine the equilibrium position of the system (student plus springs). In question 2, you are asked how far the ball would fall before it reverses direction. This is not the same as the equilibrium position and would in fact be twice as far from the initial position as the equilibrium position.
Each method, used correctly, should give you the right answer to each question. If you were to post your solutions, we could comment on where you went wrong.
 
  • #3
tnich said:
Each method, used correctly, should give you the right answer to each question.
But... in 1, it is a static arrangement. A conservation law is only directly useful when something has changed.
Conversely, in 2, it is not obvious how to find the force when the velocity is zero.
To apply energy in 1 or Hooke in 2 seems to require performing the conversion, i.e. integrating Hooke or differentiating energy.
 
  • #4
Any reason you can't turn 1) into a dynamic situation and equate the PE lost by the man with the PE gained by the spring...

mgh=0.5kh^2

Solve for h.
 
  • #5
CWatters said:
Any reason you can't turn 1) into a dynamic situation and equate the PE lost by the man with the PE gained by the spring...
Yes, there is a reason. That will give twice the correct answer, as @keroberous found and @tnich explained.
 
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Related to Hooke's Law vs. Conservation of Energy

What is Hooke's Law?

Hooke's Law is a principle in physics that describes the relationship between the force applied to an elastic object and the resulting deformation of that object. It states that the force applied is directly proportional to the amount of stretch or compression of the object, as long as the object remains within its elastic limit.

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 only transformed from one form to another. This means that the total amount of energy in a closed system remains constant, even as it changes forms.

How are Hooke's Law and Conservation of Energy related?

Hooke's Law and Conservation of Energy are related in that both principles describe the behavior of physical systems. Hooke's Law explains how elastic objects respond to external forces, while Conservation of Energy explains how energy is transformed and conserved within a system.

Can Hooke's Law be used to violate the Conservation of Energy?

No, Hooke's Law cannot be used to violate the Conservation of Energy. This is because Hooke's Law only applies to elastic objects within their elastic limit, and the deformation of these objects does not result in a loss or gain of energy. Therefore, the total energy within a closed system remains constant.

Are there any exceptions to Hooke's Law or the Conservation of Energy?

Yes, there are some cases where Hooke's Law or the Conservation of Energy may not apply. For example, Hooke's Law does not apply to objects that have reached their yield point and experience permanent deformation. Additionally, the Conservation of Energy may not hold true in systems with external forces or processes that are not taken into account.

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