There are two jack screw assemblies both with an ISO Metric Trapezoidal $\text{'}$ Tr

$163\text{'}$ screw, one single$-$thread and the other double$-$thread. Each assembly

raises and then lowers a load of $450$ kg $.$

A ball$-$thrust bearing forms part of each assembly and therefore collar friction

is negligible.

The coefficient of sliding friction for both screws is $0\mathrm{.}08.$

$($a$)$ For each assembly, determine:

$($i$)$ the pitch, pitch diameter, lead and lead angle;

$($ii$)$ the running torque required to lower the load;

$($iii$)$ whether they are back$-$driveable or self$-$locking?

$($b$)$ Which assembly is the most efficient in lifting?

$($c$)$ One way of ensuring that a lifting device can be made to self$-$lock is

to use a thrust collar rather than a ball$-$thrust bearing. For any of the

assemblies found to be back$-$driveable in $($a$)($iii$),$ what is the

minimum collar diameter required to ensure it$/$they would self$-$lock?

$($Assume the same value of friction coefficient of $0\mathrm{.}08$ in all situations$).$