(Spring constant) I can't figure out what I did wrong.

In summary, the problem involves finding the displacement and spring constant of a series of two vertically hung springs with spring constants k1 and k2, with a mass m attached to the end. The displacement is determined to be x1+x2, and the spring constant is found using the definition of spring constant, k=F/Δx=mg/(x1+x2). The attempted solution using conservation of mechanical energy is incorrect as it does not account for the additional external work done to stop the oscillation.
  • #1
Winvoker555

Homework Statement


Two springs which have spring constant of k1 and k2 respectively are vertically hung in a series. Then, a mass m is attached to the end. Find the displacement and the spring constant of this series.

Homework Equations

The Attempt at a Solution


I got the displacement x1+x2 right (which is equal to mg/k1+mg/k2).
Then, I solved for the series constant by using the conservation of energy ,and let the two springs be my system.
ΔUs = Wext
ΔUs = mg⋅(x1+x2)
k(x1+x2)2 / 2 = mg(x1 + x2)
k(x1+x2) / 2 = mg
k = 2mg/(x1 + x2)

Since x1+x2 = mg / k1+mg / k2) ,then:
k = 2mg / (mg / k1 + mg / k2)
k = (2k1k2) / (k1+k2)

I checked the solution and this is not correct (the correct answer is (k1k2) / (k1+k2) );also, I know that the formula for spring constant of a series of two springs is 1 / kseries = 1 / k1 + 1 / k2. However, I can't figure out what I did wrong and what caused the extra 2 factor in the numerator.
 
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  • #2
Winvoker555 said:

Homework Statement


Two springs which have spring constant of k1 and k2 respectively are vertically hung in a series. Then, a mass m is attached to the end. Find the displacement and the spring constant of this series.

Homework Equations

The Attempt at a Solution


I got the displacement x1+x2 right (which is equal to mg/k1+mg/k2).
Then, I solved for the series constant by using the conservation of energy ,and let the two springs be my system.
ΔUs = Wext
ΔUs = mg⋅(x1+x2)
k(x1+x2)2 / 2 = mg(x1 + x2)
k(x1+x2) / 2 = mg
k = 2mg/(x1 + x2)

Since x1+x2 = mg / k1+mg / k2) ,then:
k = 2mg / (mg / k1 + mg / k2)
k = (2k1k2) / (k1+k2)

I checked the solution and this is not correct (the correct answer is (k1k2) / (k1+k2) );also, I know that the formula for spring constant of a series of two springs is 1 / kseries = 1 / k1 + 1 / k2. However, I can't figure out what I did wrong and what caused the extra 2 factor in the numerator.
Why do you calculate the new spring constant from conservation of mechanical energy ? If you connect a mass to the end of the new spring and let it go, both the kinetic energy and the elastic energy of the spring would change.
Use simply the definition of spring constant K=F/Δx=mg/(x1+x2).
 
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  • #3
ehild said:
Why do you calculate the new spring constant from conservation of mechanical energy ? If you connect a mass to the end of the new spring and let it go, both the kinetic energy and the elastic energy of the spring would change.
Use simply the definition of spring constant K=F/Δx=mg/(x1+x2).

Thank you very much for pointing that out. I used conservation of mechanical energy because the problem stated that the system is stationary ,and I thought that I can solve for k this way. However, I now realize that if a mass m is hung at the end of the springs, there has to be an additional force acting on it to stop the oscillation ,and that will caused additional external work that I didn't include in my equation.
 
  • #4
Winvoker555 said:
Thank you very much for pointing that out. I used conservation of mechanical energy because the problem stated that the system is stationary ,and I thought that I can solve for k this way. However, I now realize that if a mass m is hung at the end of the springs, there has to be an additional force acting on it to stop the oscillation ,and that will caused additional external work that I didn't include in my equation.
You are right now. :)
 

Related to (Spring constant) I can't figure out what I did wrong.

1. What is a spring constant?

A spring constant is a measure of the stiffness of a spring. It is represented by the letter "k" and is measured in units of force per unit distance.

2. How do you calculate the spring constant?

The spring constant can be calculated by dividing the force applied to the spring by the displacement it causes. This can be represented by the equation k = F/x, where F is the force and x is the displacement.

3. Why is my calculated spring constant different from the expected value?

There are several reasons why your calculated spring constant may differ from the expected value. These include experimental errors, variations in the spring's material or manufacturing, and the presence of external factors such as friction.

4. Can the spring constant change?

Yes, the spring constant can change. It can be affected by factors such as temperature, strain, and the material properties of the spring. It can also be altered by manipulating the spring (e.g. stretching or compressing it).

5. How does the spring constant affect the behavior of a spring?

The spring constant determines how easily a spring can be stretched or compressed. A higher spring constant means the spring will require more force to stretch or compress, while a lower spring constant means it will be easier to stretch or compress. It also affects the frequency of oscillation of a spring system.

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