$($a$)$ Define compaction and state its objectives.

$[2]$

$($b$)$ Compaction tests using the $2\mathrm{.}5kg$ rammer $($BS$1377)$ on a sample of soil produce the results given in Table Q$4\mathrm{.}1$:

Table Q$4\mathrm{.}1$

$\backslash$table$[[$Bulk density $(M\frac{g}{m^3}),2\mathrm{.}123,2\mathrm{.}141,2\mathrm{.}156,2\mathrm{.}106,2\mathrm{.}067,2\mathrm{.}024],[$Water content $(\%),15\mathrm{.}40,14\mathrm{.}34,13\mathrm{.}20,11\mathrm{.}85,10\mathrm{.}94,9\mathrm{.}79]]$

$($i$)$ Calculate the dry density at each water content and, using the graph paper provided, plot the compaction curve $({}_d$:$w).$ Include the scale and title of each axis on your plot.

$[10]$

$($ii$)$ Use the plot from $($i$)$ to determine the optimum water content and maximum dry density for the soil.

$2$

$($iii$)$ Add the $0$ and $4\%$ air voids lines$*$ to the plot, assuming ${}_w=1M\frac{g}{m^3},G_s=2\mathrm{.}65,$ and using ${}_d=G_s{}_w\frac{1-A_r}{1+w{G}_s}.$ Comment on how well the objectives of compaction have been achieved at the optimum. $(*$ it will be sufficient to locate two points on each line, at water contents of II and $15\%,$ in order to draw them.$)$

$[5]$

$($c$)$ A cylindrical sample of soil is taken from the compaction mould after one of the tests reported in Table Q$4\mathrm{.}1$ above has been conducted. The sample has a diameter of $38mm,$ a length of $76mm,$ and a total mass of $185\mathrm{.}83g.$ After oven drying the sample has a mass of $164\mathrm{.}16g.$ Taking the same specific gravity of solids $G_s$ assumed in part $($b$)($iii$),$ determine the:

$($i$)$ water content $(w\%,$

$[1]$

$($ii$)$ porosity $($n$),$ and

$[1]$

$($iii$)$ air voids ratio $(A_r\%).$

$[1]$

Draw a phase $($block$)$ diagram, indicating all relevant volumes and masses.

$[3]$