# Spring system related question [closed]

I was trying to do some elementary physics practice questions and came across this one that got me so confused...

Here you see this spring system and I wanted to show that the the value of $k_1$ to ensure that a force $F$ displaces the spring system by a distance $x$ is given by $$k_1 = \frac{Fk_2}{2k_2x-2F}$$

I'm just so confused right now and don't know what to do.

I tried to get the $x$ distance equation in this case $$x = F(k_2 + 2k_1)/2k_1k_2$$

And now I'm just out of wits. Hopefully someone can shed some lights on this.

• Let $\:n\:$ springs with constants $\:k_{1},k_{2},\cdots,k_{n}$. Try to prove that the equivalent constant $\:k_{\rm{total}}\:$ is \begin{align} k_{\rm{total}}^{\rm{parallel}} & =k_{1}+k_{2}+\cdots+k_{n} \quad \text{(parallel arrangement)} \tag{01}\\ &\\ \dfrac{1}{k_{\rm{total}}^{\rm{series}}} & =\dfrac{1}{k_{1}}+\dfrac{1}{k_{2}}+\cdots+\dfrac{1}{k_{n}}\quad \text{(series arrangement)} \tag{02} \end{align} – Frobenius Oct 4 '16 at 7:54

just solve for $k_1$ from the above equation you got. you are there!
• no tricks here. its all very straightforward. the equivalent spring constant is $2k_2$||$k_1$. put F=$2k_2$||$k_1$x and then simply solve for $k_1$ – Prasad Mani Oct 4 '16 at 6:20
• $2k_2$ and $k_1$ are in parallel combination. First the two springs of $k_1$ next to each other serve to add their spring constants. Its value is then $2k_1$($k_1$+$k_1$). Then the spring of $k_2$ which is alongside the two springs acts in such a way that the equivalent spring constant now becomes the parallel combination of $2k_2$ and $k_1$. That is, $k_{resultant}$=$\frac{(2k_1)(k_2)}{2k_1+k_2}$ – Prasad Mani Oct 4 '16 at 6:25
• oh yeah sorry. its $2k_1$. let me edit that – Prasad Mani Oct 4 '16 at 6:27
You need to consider two extensions, the extensions of the top springs, and the extensions of the bottom spring. Call them $x_1$ and $x_2$ then $x=x_1 + x_2$. Then solve for the values of $x_1$ and $x_2$ given a force $F$. You also need to consider the forces on the central beam.