$4\mathrm{.}2$ The single body shown has a mass of $8\mathrm{.}0$ kg and a moment of inertia of $0\mathrm{.}4kg.m^2.$ In the

shown configuration the coordinates and velocity of the body are:

$r=\{[2\mathrm{.}0,1\mathrm{.}6]{\}}^{\text{'}}m,=30,r^{}=\{[0\mathrm{.}15,-0\mathrm{.}25]{\}}^{\text{'}}\frac{m}{s},{}^{}=0\mathrm{.}6ra\frac{d}{s}$

Gravity acts on the system. A spring is connected between point $P$ and $A,$ and a damper

between points $Q$ and $B.$ The coordinates of these points are:

$s^P=\{[-1\mathrm{.}6,0\mathrm{.}8]{\}}^{\text{'}},s^Q=\{[2\mathrm{.}0,0]{\}}^{\text{'}},r^A=\{[-2\mathrm{.}0,3\mathrm{.}0]{\}}^{\text{'}},r^B=\{[4\mathrm{.}0,0]{\}}^{\text{'}}m$

The spring and the damper have the following characteristics:

$k=15\mathrm{.}0\frac{N}{m},{?}^{0}L=1\mathrm{.}0m,d_c=3\mathrm{.}0N.\frac{s}{m}$

$($a$)$ Construct the mass matrix for the rod

$($b$)$ Determine the force of the spring

$($c$)$ Determine the force of the damper

$($d$)$ Construct the $31$ array of applied forces $($first show the contribution of the

gravitational force, the force of the spring, and the force of the damper to the array of

applied forces separately before adding them together$).$

$($e$)$ Construct and solve the equations of motion for the accelerations