Find Spring Constant for Two Springs in Series

[StephenPrivitera]

Two springs are in series. You know the constants for each individual spring. How do you find the spring constant for the spring system?

|-OOOOO---o---OOOOOOOOOO-| --->F
In the middle of the spring system (where the "o" is) the net force is zero and we have
k1x1=k2x2 disregarding sign
F=k3x3
x3 =x2+x1
F=k3(x2+x1)
F=k3(k1x1/k2+x1)
F=k3*x1/k2*(k1+k2)
but F=k1x1+k2x2
F=2k1x1
2k1x1=k3*x1/k2*(k1+k2)
2k1k2=k3(k1+k2)
k3=2k1k2/(k1+k2)

Is this right?

 

[heumpje]

Looks OK.
Now try to find a similar expression for N springs..:)

 

[Integral]

Looks to me like you have an extra factor of 2 in there. Springs in series add as Resistors in parallel. So

1/K_T = 1/K₁+ 1/K₂

OR


K_T= (K₁+K₂)/(K₁K₂)

Also Springs in parallel add as resistors in series.

K_T=K₁+K₂

I do not have time to do the derivation now. Should be able to post it later to day. A key is that in the series case the force seen by each spring is equal, while in the parallel case the extensions are equal.

 

[StephenPrivitera]

Originally posted by Integral
Looks to me like you have an extra factor of 2 in there. Springs in series add as Resistors in parallel. So

1/K_T = 1/K₁+ 1/K₂

Ah, I see my mistake. I have F=k1x1+k2x2 but F=k2x1=k1x1. I need to be more careful.

 

[Integral]

OK, that looks better!
http://home.comcast.net/~rossgr1/Math/Spring_constant.PDF is my derivation. As long as I was at it I did the case for parallel springs also.

BTW: it would be trivial to extend this to multiple springs.

 

[StephenPrivitera]

Thank you, Integral. I have put your work into my physics notebook (hope you don't mind).