(a) For the elastically supported mass on a vertically moving base shown in Figure 5.37 derive the governing equation of motion first using d'Alembert's principle, and then using Lagrange's equation. Discuss the effect of base motion and show that it is equivalent to a force on mass \(m\) equal to \(k A \sin \omega t\).

(b) Repeat the problem where viscous damping exists due to the motion of the mass through the surrounding medium. Assume a damping constant denoted by \(c\).

www m x = A sin oot Static equilibrium 9 Figure 5.37: Elastically supported mass on a moving base.