A particle is attached to one end of an unstretched Hooke's-law spring of force-constant \(k\). The other end of the spring is fixed in place. If now the particle is pulled so the spring is stretched by a distance \(x\), the potential energy of the particle is \(U=(1 / 2) k x^{2}\).

(a) Now suppose there are two springs with the same force constant that are laid end-to-end in the \(y\) direction, with a particle attached between them. The other ends of the springs are fixed in place. Now the particle is pulled in the transverse direction a distance \(x\). Find its potential energy \(U(x)\).

(b) \(U(x)\) is proportional to what power of \(x\) in the limit of small \(x\), and to what power of \(x\) in the limit of large \(x\) ?