I have some trouble with springs in series [duplicate]

Question

I am starting with problems of combined strings. I think the basic ones are strings in parallel and in series. For me it´s easy to understand the formula for the parallel but don´t understand how the formula for the series configuration is derived.

If we have two spring in series and a block hanging the forces are the ones shown above.

I´ve seen several derivations and they always say that $k_1x_1= k_2x_2$, I know that it has to be with the forces but can´t see it. The process after that is clear

Answer 1

The force sping 1 exerts on spring 2 at their point of connection is $k_1 x_1$ (upward) if $x_1$ is the extension. The force spring 2 exerts on spring 1 is $k_2 x_2$ (downward). By Newton's third law these forces are equal and oposite. So $k_1 x_1= k_2 x_2$.

Answer 2

If the springs are assumed to be massless, then the tensions in all of the springs connected in series have to be equal. In your free body diagram, for instance, the tension of both springs is equal to $F$. This is a consequence of Newton's third law.

We know that the tension within a spring goes as $F=kx$. If the spring constants are $k_{1}$ and $k_{2}$ respectively, we can immediately write down $F = k_{1}x_{1} = k_{2}x_{2}$. Finally, the total extension is the sum of the individual extensions. Putting this together,

$x_{t}=x_{1}+x_{2} \implies \frac{F}{k_{t}}=\frac{F}{k_{1}}+\frac{F}{k_{2}} \implies \frac{1}{k_{t}} = \frac{1}{k_{1}}+\frac{1}{k_{2}}$

Answer 3

Here, it is assumed that the tension throughout the spring system is the same and is equal to force F as shown in the question. If the connected strings are of different materials, their K value would be different and hence the different elongations, but they always satisfy the following equation.

$$ F = k_1x_1= k_2x_2 $$