Multiple Springs (Parallel and Series)

Now that you are comfortable with dealing with spring, let's deal with multiple springs! One natural question is $-$ if I have two springs, how will it make the mass behave in place on one spring? As physicists or engineers, what we sometimes ask is this: if I have a two-spring system, it would behave the same way as if I had one spring that "combined" the two. There are two fundamental ways to combine two springs and hold one mass. The two are shown below.

The configuration on the left is called parallel and the one of the right is called series; both should be intuitive.

First, let us consider the parallel case. Let us say one of the springs has spring constant $k_1$ and the other one has spring constant $k_2$. What should the spring constant for the combined spring be? By the way, we will call this combine spring constant the effective spring constant.

Correct! So, let us say both springs are displaced $\Delta x$. Then, we know that, if the effective spring constant of the system is $k$, we have that the total force is $k \cdot \Delta x$. However, it is also the sum of the two spring forces; i.e. $k_1 \cdot \Delta x + k_2 \cdot \Delta x$. Hence, we have $$k \cdot \Delta x = k_1 \cdot \Delta x + k_2 \cdot \Delta x \implies k = k_1 + k_2.$$

Now, the applied forces are the same, so $k_1 \cdot \Delta x_1 = k_2 \cdot \Delta x_2$, where $x_i$ is the displacement of the spring with spring constant $k_i$.

Now, it should be clear that the total displacement of the system is the sum of the displacements of each spring (because they are in a straight line in the same direction, so the displacements just add). Hence, we have $x = x_1 + x_2$, where $x$ is the displacement of the mass/spring-system.

Let $F$ be the common force. Then, we get $$x = \frac {F}{k_1} + \frac {F}{k_2} = F \cdot \left( \frac {1}{k_1} + \frac {1}{k_2} \right),$$ so if $F=kx$, we have $$k = \frac {1}{\frac {1}{k_1} + \frac {1}{k_2}} = \frac {k_1 k_2}{k_1 + k_2}.$$This is nice because it is symmetric as it should be; i.e. the order of the springs does not matter!