$1.$ A horizontal beam is simply supported at A and D and is loaded as shown in Figure $1.$ The beam has a square cross section with dimensions $\backslash ($ b$=$h $\backslash ).$

$($a$)$ Determine the vertical reactions at the supports A and D $.$

$(8$ marks$)[\backslash (3\mathrm{.}58\backslash$mathrm$\{$kN$\},3\mathrm{.}42\backslash$mathrm$\{$kN$\}]\backslash )$

$($b$)$ If the allowable deflection at mid$-$span $($i$.$e$.\backslash ($ X$=1\mathrm{.}2\backslash$mathrm$\{$~m$\}\backslash ))$ is $0\mathrm{.}7$ mm downwards, determine the cross section dimension of the beam needed. Use $\backslash (\backslash$mathrm$\{$E$\}=200\backslash$mathrm$\{$GPa$\}\backslash ).$

$[112$mm$]$

$(22$ marks$)$

$2.$ A $\backslash (40^\{\backslash$circ$\}\backslash )$ V$-$thread power screw used in lifting a load axially. A motor used to rotate a nut provides a power of $1\mathrm{.}5$ KW $,$ turning it at a $\backslash (300\backslash$mathrm$\{$rev$\}/\backslash$mathrm$\{$min$\}\backslash ).$ The screw has mean diameter of $35$ mm and a pitch of $8$ mm $.$ The nut sits on a collar having outer diameter of $90$ mm and inside diameter of $70$ mm $.$ The coefficients of friction are $0\mathrm{.}15$ and $0\mathrm{.}18$ for the threads and collar respectively. Determine:

$($a$)$ the helix and friction angles of the screw thread, $[4\mathrm{.}16,9\mathrm{.}07]$

$(4$ marks$)$

$($b$)$ the total torque that is being applied to lift the load, $[47\mathrm{.}75$ Nm$]$

$(4$ marks$)$

$($c$)$ the torque required to overcome screw thread friction in terms of the load, $[0\mathrm{.}00411$ W Nm$]$

$(5$ marks$)$

$($d$)$ the torque required to overcome collar friction in terms of the load,

$[0\mathrm{.}0072$ W Nm$]$

$(6$ marks$)$

$($e$)$ the maximum load that can be lifted, $\backslash ([4221\mathrm{.}93\backslash$mathrm$\{$~N$\}]\backslash )$

$(5$ marks$)$

$($f$)$ the linear speed $($in $\backslash (\backslash$mathrm$\{$mm$\}/\backslash$mathrm$\{$s$\}\backslash ))$ at which this load is being lifted, and

$[40$ mm$/$s$]$

$(2$ marks$)$

$($g$)$ the combined efficiency of the screw and the collar.