# B Why doesn't the Spring Force change on splitting?

#### navneet9431

Gold Member
Suppose a spring of spring constant=K and Length=L is split into two parts L1 and L2, with spring constants K1 and K2 respectively.

Then,why is it such that the spring force F=K1*L1=K2*L2=K*L?

Please give an intuitive explanation of why the spring force doesn't change?

I will be thankful for help!

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#### Delta2

Homework Helper
Gold Member
If I understand correctly, you asking why all those forces are equal in the case we have two springs connected in series.

Lets focus on spring with constant $K_2$. To this spring are connected the spring with constant $K_1$ and the block (whose displacement is $L=L_1+L_2$ from the equilibrium position)
We can view the spring with constant $K_1$ as another hypothetical block connected to spring $K_2$. Hence the force that the spring $K_2$ applies to this hypothetical block is by hook's law equal to $$F_{21}=K_2L_2 (1)$$.
By Newton's 3rd law the hypothetical block applies to the spring $K_2$ a force opposite and equal $$F_{12}=F_{21} (2)$$.

Now we focus on spring $K_1$. To this spring is connected the spring $K_2$ which also can be viewed as a hypothetical block connected to spring $K_1$. The force applied from spring $K_1$ to this hypothetical block is essentially the force $F_{12}$ and by hook's law it is equal $$F_{12}=K_1L_1 (3)$$
By combining (1) (2) and (3) we get that $K_1L_1=K_2L_2$.

Maybe my explanation is not very intuitive but I don't see any other way how we can prove it, without viewing the springs as hypothetical blocks and also have to use Newtons 3rd Law.

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