$($a$)$ Define 'instantaneous centre' and show that, if two rigid bodies have plane

motion relative to a third, to which they are connected, the three instantancous

centres lie on a straight line.

$(4$ marks$)$

$($b$)$ Figure Q$1$ shows one of the four$-$bar linkages used in a stitching machine. For

the position shown, find the angular velocities of the coupler link $AB$ and the

output line $B_{0}B$ when the input line $A_0$A rotates at a constant speed of $500re\frac{v}{m}in.$

Also calculate the angular velocity of the output link $B{B}_0$ at this instant. The lengths of

the links are $AB=A_{0}B=100mm$;$A_{0}A=40mm$;$B_{0}B=50mm.$

$(10$ marks$)$