$2.$ A square thread power screw is to carry a load of $3$ KN $.$ The screw is double threaded, has $4$ mm pitch, $32$ mm pitch diameter and mean diameter and $80$ mm collar diameter.

$($a$)$ Calculate the minimum value of friction coefficient if the screw must be self$-$locked in the absence of collar friction.

$($b$)$ Assume that friction coefficient is present in all surfaces of contact. Determine the torque and mechanical advantage if the load must be raised in absence of collar friction.

$($c$)$ Find the torque required to overcome collar friction when the load is lowered.