The curve \(C\) has parametric equations \(x=\frac{3}{2} t^{2}, y=t^{3}\), for \(0 \leqslant t \leqslant 2\). Find the arc length of \(C\).

Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x=6\).