Question #42787

An infinite mechanical ladder network is constructed using massless springs, analogous to an infinite capacitor ladder. The network consists of nodes numbered $n = 0, 1, 2, \dots, \infty$. For each $n \ge 1$, a series spring of stiffness $k_1 = 6k$ connects node $n-1$ to node $n$, and a shunt spring of stiffness $k_2 = k$ connects node $n$ to a fixed rigid wall. A constant static force $F$ is applied at node 0, and the system is allowed to reach static equilibrium. Let $U_{s1}$ be the elastic potential energy stored in the first series spring (between node 0 and node 1), and $U_{p,\infty}$ be the total elastic potential energy stored in all the shunt springs combined. What is the ratio $\frac{U_{s1}}{U_{p,\infty}}$?