$1\mathrm{.}86.$ Use Lagrange's formulation to calculate the equation of motion and the natural fre$-$

quency of the system of Figure P$1\mathrm{.}86.$ This figure represents a simplified model of a jet

engine mounted to a wing through a mechanism which acts as a spring of stiffness $k$

and mass $m_s.$ Assume the engine has inertia $J$ and mass $m$ and that the rotation of the

engine is related to the vertical displacement of the engine, $x(t)$ by the "radius" $r_0($i$.$e$.,$

$x=r_0).$