Hooke's law and the Coulomb force

In summary, an unstrained horizontal spring with a length of 0.32 m and a spring constant of 220 N/m has two small charged objects attached to each end with equal magnitude charges. The spring stretches 0.020 m due to the charges. Using Coulomb's law and Hooke's law, the magnitude of the charges can be determined to be 5.16 micro Coulombs. However, an external force or kinetic energy must be considered in the analysis to account for the work done in guiding the charges to the equilibrium position.
  • #1
Ampere
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Homework Statement



An unstrained horizontal spring has a length of 0.32 m and a spring constant of 220 N/m. Two small charged objects are attached to this spring, one at each end. the charges on the objects have equal magnitudes. Because of these charges, the spring stretches 0.020 m, relative to its unstrained length. Determine the magnitude of the charges.

Homework Equations



Hooke's law: F=-kx
Spring potential energy; E = 1/2kx^2
Coulomb's law; F=kq1q2/r^2
Voltage; V = kq/r

The Attempt at a Solution



I know energy is conserved so ΔPE + q*ΔV = 0

which gave me 0.5*220*0.02^2 = (8.99*10^9)*q^2*(1/0.34 - 1/0.32)

And q = 5.16 micro Coulombs.

But why do I get a different answer if I say that the forces are equal in equilibrium, i.e.

kx = (8.99*10^9)q^2/r^2
(220)(0.02)=(8.99*10^9)(q^2)/(0.34^2)?
 
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  • #2
What do you mean, 'energy is conserved'?

Your second method, equating spring force with Coulomb force, is correct.
 
  • #3
rude man said:
What do you mean, 'energy is conserved'?

Your second method, equating spring force with Coulomb force, is correct.
To elaborate on rude man's answer, if you are going to use energy considerations, you need to recognize that in the initial state, you need to have an externally applied force to hold the charges in place. As you ease up on this force, the spring stretches and the charges get farther apart. Your missing energy is the work done on the external agent as you ease up on the force.
 
  • #4
"Your missing energy is the work done on the external agent as you ease up on the force."

I don't understand this - any physics problem would require an "external agent" to set it up and/or hold it in place. Consider block-and-pulley problems. Before releasing the block, you would need to hold at least one of them in place, but there is no external agent in the energy equation.

It requires no work to hold the charges in place. If I simply release the charges, energy should be conserved from that point on. Aren't the spring and electric forces conservative? Yes, someone would have done work to push the charges together from infinity, but none of that work would be recovered if that person simply released the charges all at once.

The reason why I don't like using the forces is that the problem doesn't really say the spring is in equilibrium, just that the spring stretches. (Maybe it will stretch more before it's in equilibrium.) I would like to avoid making the equilibrium assumption if possible.
 
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  • #5
Ampere said:
"Your missing energy is the work done on the external agent as you ease up on the force."

I don't understand this - any physics problem would require an "external agent" to set it up and/or hold it in place. Consider block-and-pulley problems. Before releasing the block, you would need to hold at least one of them in place, but there is no external agent in the energy equation.

It requires no work to hold the charges in place. If I simply release the charges, energy should be conserved from that point on. Aren't the spring and electric forces conservative? Yes, someone would have done work to push the charges together from infinity, but none of that work would be recovered if that person simply released the charges all at once.

The reason why I don't like using the forces is that the problem doesn't really say the spring is in equilibrium, just that the spring stretches. (Maybe it will stretch more before it's in equilibrium.) I would like to avoid making the equilibrium assumption if possible.

In the examples you cited, there is also kinetic energy produced. The charged bodies have mass, so, if you release them suddenly, and no external force is subsequently being applied, the charged masses will oscillate about the equilibrium position. If you are unhappy with applying an external force, I have no problem with replacing the work provided by this force with kinetic energy of the charged masses. However, then the 0.02 meter displacement will only represent the equilibrium position, and you won't have enough information to ascertain the magnitudes of the charges. When the two charges are at the equilibrium location, the missing energy in your analysis will be equal to the maximum kinetic energy experienced by the charges over the oscillatory cycle.
 
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  • #6
Okay that makes sense.

What you're saying is that, to prevent oscillation, someone must have "guided" the charges to the 2cm equilibrium position, and work was done as they did that.

Thanks for the help. I hadn't thought of it that way.
 

1. What is Hooke's law?

Hooke's law is a principle in physics that states the force needed to extend or compress a spring by some distance is directly proportional to that distance. This relationship can be expressed mathematically as F = -kx, where F is the force, k is the spring constant, and x is the distance.

2. How is Hooke's law related to the Coulomb force?

Hooke's law and the Coulomb force are both examples of conservative forces, meaning that they can be described by a potential energy function. In the case of Hooke's law, the potential energy is related to the displacement of the spring, while in the Coulomb force, it is related to the distance between two charged particles.

3. What is the Coulomb force?

The Coulomb force is the electrostatic force of attraction or repulsion between two charged particles. It is proportional to the product of the charges and inversely proportional to the square of the distance between them, and can be expressed mathematically as F = kq1q2/r2, where F is the force, k is the Coulomb constant, q1 and q2 are the charges, and r is the distance between them.

4. How are Hooke's law and the Coulomb force used in practical applications?

Hooke's law is commonly used in the design of springs, such as those in car suspension systems or mattress coils. The Coulomb force is fundamental to understanding and predicting the behavior of charged particles, and is used in a variety of fields including electronics, chemistry, and astronomy.

5. What are the limitations of Hooke's law and the Coulomb force?

Hooke's law is only applicable to objects that exhibit linear elastic behavior, meaning that they return to their original shape when the force is removed. The Coulomb force is limited to point charges and does not take into account the effects of quantum mechanics. In certain situations, such as when dealing with highly charged particles or at very small distances, other forces may need to be considered.

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