The potential energy scalar product for a uniform bar is defined as

\[
(f, g)_{v}=\int_{0}^{L} E A f(x) \frac{d^{2} g}{d x^{2}} d x
\]

Consider the cases where

(a) the bar is fixed at \(x=0\) and free at \(x=L\) and

(b) the bar is fixed at \(x=0\) and attached to a linear spring of stiffness \(k\) at \(x=L\). Discuss, in each case, the implication of requiring \(f(x)\) and \(g(x)\) to satisfy only the geometric boundary conditions.