System # $3$

System #$3$ is reported in Figure $3.$ It is constituted by a rigid uniform bar of total mass equal to $m.$ The

bar is analyzed by using the generalized coordinate var$$ depicted in Figure $3.G$ is the center of mass of the

bar and is located in the middle of the bar, which has length $L.$ The following constraints are in place:

A hinge is present at point $O,$ which is located at a distance of $\frac{L}{4}$ from one end $($see Figure $3).$

A dashpot with constant $c$ is placed at one end $($see Figure $3).$ The dashpot is assumed to stay

horizontal during the motion. Part $3$

Consider system # $3.$ Please perform the following tasks:

For a value of var$$ that cannot be considered small, present the free body diagram with the inclusions

of applied forces, reactions, intertial forces, and moment of inertial forces. Task # $3\mathrm{.}1$

Take the free body diagram and write the equation of motion. Task # $3\mathrm{.}2$

Using the equilibrium conditions $($under dynamic conditions$),$ express the $2$ translational reactions

at the hinge O as a function of var$$ and its time derivatives. Task # $3\mathrm{.}3$

Assume var$$ is small. Linearize the equation of motion about the static equilibrium identified by var$=0$

and var${}^{}=0.$ Task# $3\mathrm{.}4$