The two link, pin$-$jointed $($O$,$ B$),1$ DOF mechanism shown is operated in a horizontal plane with link $($#$2)$ having a constant input angular velocity $({}_2)$ driven by an unknown torque $(T_2)$ and supporting a constant, horizontal applied load $(F_{\text{l}\text{o}\text{a}\text{d}})$ at the tip $(G_3)$ of link $($#$3)$ which rides in a smooth horizontal slot. Links $($#$2)$ & $($#$3)$ are both built of rigid, ideal rods of length L$.$ Link $($#$2)$ has mass $(m_2)$ and rotational inertia $(I_{28})$ concentrated at end $(G_2).$ Link $($#$3)$ consists of a particle mass $(m_3)$ of negligible dimensions fixed to end $(G_3).$ Assume all joints and contact surfaces are smooth.

With the vector loop as shown, the VLE constraint is as follows:

$R_{{}_2,{}_3,R_A}=R_2{e}^{i{}_2}+R_3{e}^{i{}_3}-R_A{e}^{i0}={}_{?}$

Input: ${}_2$ Known$/$Fixed: $R_2=R_3=L,{}_A=0$ Unknown$/$Variable: ${}_3,R_A$

The symmetry of the mechanism allows for quick resolution of the unknowns and the corresponding $1^{st}$&$2^{nd}$ Order Kinematic Coefficients as follows:

Position: $,{}_3=-{}_2$

$R_A=2$Lcos${}_2$

$f_A=-2$Lsin${}_2$

$f_A^{\text{'}}=-2$Lcos${}_2.$

$1^{st}$ Order $KC$: $h_3=-1$

$2^{nd}$ Order $KC$: $h_3^{\text{'}}=0$

i$)$ Construct appropriate Free Body Diagrams $($FBDs$)$ of links $($#$2)$ & $($#$3).$

Write the dynamics constraint equations $(6$ in total$)$ and formulate the Inverse Dynamics Problem as a matrix equation of the form $[A]\{x\}=\{b\}.$ Clearly specify your A coefficient matrix, your $\{x\}$ unknowns vector, and your $\{b\}$ constants vector.

ii$)$ Explain which of the four Power Equation terms are needed $($or not$)$ for finding the unknown Torque $(T_2)?$ Why? Formulate the specific, terms that are required and solve for $T_2.($Do not rederive the Power Equation$)$

$P=d\frac{T}{d}t+d\frac{U}{d}t-P_f$