# 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.

"Why doesn't the Spring Force change on splitting?"

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