P$2.[12$ marks total$]$ Two cylindrical blocks A and B are connected by the pulley system as shown in the diagram. Cylinder $\backslash ($ A $\backslash )$ has a mass of $8$ kg and cylinder $\backslash ($ B $\backslash )$ has a mass of $6$ kg $.$ Neglect the mass of the pulleys and cords.

$($a$)[3$ marks$]$ Draw a datum line on the diagram. Further indicate on the diagram the direction and reference distances for appropriate lengths of cord which connect the two cylinders, $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ and $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$B$\}\}\backslash ),$ with respect to this datum line.

$($b$)[3$ marks$]$ Write down an expression for the length of the cord with respect to $\backslash (\backslash$mathrm$\{$S$\}\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ and SB from part $($a$),$ and therefore further write down an equation for the relationship between the acceleration of block A and $\backslash (\backslash$mathrm$\{$B$\},\backslash$mathrm$\{$a$\}\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ and $\backslash (\backslash$mathrm$\{$a$\}\_\{\backslash$mathrm$\{$B$\}\}\backslash ),$ respectively, in the $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ and $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$B$\}\}\backslash )$ coordinate systems you drew in part $($a$).$

$($c$)[4$ marks$]$ Further write down two equations of motion $[$Newton$\text{'}$s $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ Law$]$ for each of the cylindrical masses, and solve the equations to calculate the magnitude of the tension in the cord, $\backslash ($ T $\backslash ),$ in Newtons.

$($d$)[2$ marks$]$ Finally, if the system is released from rest, calculate the speed of cylinder $\backslash ($ A $\backslash )$ after $4$ seconds. You may assume that the motion of the pulleys is not impeded by hitting any of the mounts during this time.