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

[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.
ΔU_s = W_ext
ΔU_s = 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 / k_series = 1 / k₁ + 1 / k₂. However, I can't figure out what I did wrong and what caused the extra 2 factor in the numerator.

 

[ehild]

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.
ΔU_s = W_ext
ΔU_s = 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 / k_series = 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).

 

[Winvoker555]

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.

 

[ehild]

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. :)