Question $3(40)$

For the following slider$-$crank mechanism:

$($a$)$ Write down the scalar velocity loop equations in terms of the dimensions and the position variables of the mechanism.

$($b$)$ Determine the analytical expressions for the velocity variables ${}_{13},s_{14},$ in terms of input velocity ${}_{12},$ position variables and fixed dimensions

$($c$)$ Write down the scalar acceleration loop equations in terms of the dimensions, the position and the velocity variables of the mechanism.

$($d$)$ Determine the analytical expressions for the acceleration variables ${}_{13},s_{14}^{\text{'}\text{'}}$ in terms of input acceleration ${}_{12},$ velocity variables, position variables and fixed dimensions.

$($e$)$ For given dimension and constant angular speed of ${}_{12}=5ra\frac{d}{?}$; determine velocity and acceleration of piston and $($:$s_{14}^{**}$