Let \(w_{1}(x), w_{2}(x), w_{3}(x), w_{4}(x)\) be linearly independent polynomials of degree four or less that satisfy the geometric boundary conditions for a bar fixed at \(x=0\) and attached to a spring of stiffness \(k\) at \(x=L\).

(a) Determine a set of \(w_{1}(x), w_{2}(x), w_{3}(x), w_{4}(x)\).

(b) Use the assumed modes method with the functions obtained in part

(a) as trial functions and \(k L^{3} / E I=0.5\) to approximate the system's lowest natural frequencies and mode shapes.