Q$1.[6$marks$]$

Holding by a $3-$jaw chuck in a conventional lathe, a mild steel round bar of $140$ mm diameter and $200$ mm long is face$-$turned and its length gets reduced by $2$ mm $.$ In the entire facing operation, the spent time solely for cutting is $1$ min $.$ Given that the lead screw $($having single start thread$)$ driving the cross slide for transverse motion of the tool has a pitch of $5$ mm and the gear mounted on it to receive transmission from apron$-$assembly has $25$ teeth. The wormworm wheel combination on the feed rod offers a change gear ratio of $1$:$40.$ The effective transmission ratio $(u_{cg})$ from the spindle to feed rod is set for reduction of speed by $\frac{1}{3}.$ If the co$-$gear of the worm wheel in the apron mechanism has $70$ no of teeth and the intermediate gear transmitting the motion to cross slide lead$-$screw gear is $30,$ determine the surface speed of the work $($in $\frac{m}{m}in)$ and transverse feed rate of the tool $($in $m\frac{m}{r}ev)$ during the operation. Refer to the following kinematic diagram $($Figure $1)$ of the lathe being used for the operation.