Question $1[15$ marks$]$ Consider the slider$-$crank mechanism in which the crank OA rotates by an angle $$ from the inner dead centre. Let the length of the connecting rod and crank be $l$ and $r$ respectively. The $x$ be the displacement of the piston from the inner dead centre. Find the

$($a$)$ expression for $x$ in terms of $r$ and $,$ the velocity $v=\frac{dx}{(d)t}$ and acceleration of the piston $a=\frac{dv}{(d)t}.(2$ marks$)$

$($b$)$ expression for the angular acceleration of the connecting rod in terms of the angular velocity $,l,r,$ and $.(2$ marks$)$

$($c$)$ Comment for the above cases when $l>>r.(1$ mark$)$

Figure for question $1$: Slider$-$crank mechanism