# Series combination of springs

When a spring mass system is connected vertically with two massless springs in series whose spring constants are $$k_1$$ and $$k_2$$ to a block of mass $$m$$ we know that equal forces act on both the springs. Let that force during oscillations be $$F$$.

When we calculate effective spring constant $$k_s$$, why don't we say the net force acting on the system is $$2F$$?

Finding net force acting on the above system:

When the block is attached,the system attains equilibrium position through displacements $$x'_1$$ and $$x'_2$$.

At equilibrium: $$2F'=mg$$(Where $$F'$$ is magnitude of spring force initially by each spring)

So, $$k_1x'_1+k_2x'_2=mg$$ (equation 1)

When the system is pulled down it makes oscillations,now:

Total elongation be $$x$$

Elongation in spring 1 be $$x_1$$ and elongation in spring 2 be $$x_2$$.

Total spring force $$= -k_1x'_1-k_2x'_2-k_1x_1-k_2x_2$$

Total forces acting on the system $$= -k_1x'_1-k_2x'_2-k_1x_1-k_2x_2+mg = -mg-k_1x_1-k_2x_2+mg$$
(from equation 1)

So, total force $$= -k_1x_1-k_2x_2 = F_1+F_2=2F$$(as we know that both forces are equal)

So net force acting on the system is $$2F$$

The way I calculated effective spring constant is:

$$x=x_1+x_2$$

$$2F/k_s = F/k_1 + F/k_2$$

$$2/k _s = 1/k_1 +1/k_2$$

But that is not a correct equation. What's wrong in taking net force acting on system as $$2F$$.

When the springs (assumed to be massless) are hung upside down, they will have a zero extension. So in fact $$mg = k_1x+k_2x$$ when the mass is added to the system.

You wrote that the resultant force $$F_r = F_1+F_2$$ and concluded that this should be $$2F$$ as if the force in both springs were equal.

So what you should have done is write the resultant displacement $$x=x_1+x_2$$ so that $$\frac{mg}{k_s}=\frac{mg}{k_1}+\frac{mg}{k_2}$$ where $$k_s$$ is a effective spring constant. This will then give $$k_s=(\frac{1}{k_1}+\frac{1}{k_2})^{-1}$$

The springs are massless, so the tension at any point in either spring is the same value, $$F$$. For the two individual springs, with extensions $$x_1$$ and $$x_2$$, we have#

$$F = k_1x_1=k_2x_2$$

and when considered as a single spring they have an effective spring constant $$k$$ such that

$$F = k(x_1+x_2) \\\displaystyle \Rightarrow F = k\left(\frac F {k_1} + \frac F {k_2} \right) \\\displaystyle \Rightarrow \frac 1 k = \frac 1 {k_1} + \frac 1 {k_2}$$

Note that if we join two identical springs in series (so $$k_1=k_2$$ and $$x_1=x_2$$) then the spring constant of the joint spring is only half that of its two components. This is because it produces the same force $$F$$ with twice the extension of the individual springs, since $$x=2x_1$$.

ok so the first condition for the series spring system is : The spring force in the entire system is the same i.e the tension in the springs is the same

therefore
$$k_1x_1=k_2x_2$$

$$k_e(x_1+x_2)=k_1x_1$$
where $$k_e$$ is the equivalent spring constant

so if the system is vertical let gravitation force, mg, be $$F$$

hence $$x_1=F/k_1$$ ,$$x_2=F/k_2$$ and $$x_1+x_2=F/k_e$$

$$x_1+x_2=F/k_1+F/k_2=F/k_e$$

$$\Rightarrow 1/k_e=1/k_1+1/k_2$$

Now let's deal with your question, So I recommend You look into the free body diagram again. The tension in the entire system is equal to $$F$$

If u are facing troubles you can also try relating this with electricity resistor circuits. In a series connection, the current is the same through the resistors similarly in a series spring connection the tension is the same throughout. And in the case of a parallel connection the current splits in the branches but with each branch having the same potential difference, similarly, in a parallel spring system the two springs experience different tensions but extend/ contract equally

For further reference: I have some trouble with springs in series

and

https://physics.stackexchange.com/questions/311111/why-springs-in-series-experience-equal-force#:~:text=In%20series%20circuits%3B%20the%20current,force%20must%20be%20the%20

• Can you attach a free body diagram of the above case representing all the forces acting on the system? Apr 4, 2021 at 7:27