Potential energy of pair of springs. Two springs each of natural length $a$ and spring constant $C$ are fixed at points $(-a, 0)$ and $(+a, 0)$ and connected together at the other ends. In the following assume that either may expand or contract in length without buckling (see Fig. 5.19).
(a) Show that the potential energy of the system, for a displacement to $(x, y)$ of the joined ends, is
$$
\begin{aligned}
U=\frac{C}{2}\left\{\left[(x+a)^{2}+y^{2}\right]^{\frac{1}{2}}-a\right.&\}^{2} \\
&+\frac{C}{2}\left\{\left[(a-x)^{2}+y^{2}\right]^{\frac{1}{2}}-a\right\}^{2}
\end{aligned}
$$
(b) The potential energy depends on both $x$ and $y$, and we must therefore use partial differentiation to evaluate the relevant forces. Remember that the partial derivative of a function $f(x, y)$ is taken by the usual rules of differentiation according to
$$
\begin{aligned}
\frac{\partial f(x, y)}{\partial x}=\frac{d}{d x} f(x ; y=\text { const }) \\
\qquad \frac{\partial f(x, y)}{\partial y}=\frac{d}{d y} f(x=\text { const } ; y)
\end{aligned}
$$
Find the force component $F_{x}$ and show that $F_{x}=0$ for $\mathrm{r}=0$
(c) Find $F_{v}$ for $x=0$. Check the signs carefully to make sure the answer makes sense.
(d) Sketch a graph of potential energy as a function of $\mathbf{r}$ in the $x y$ plane, and find the equilibrium position.