Centre of mass with given condition

2 A uniform lamina \(A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and diameter 0.4 m , and an isosceles triangle \(A B D\) with base \(B D = 0.4 \mathrm {~m}\) and perpendicular height \(h \mathrm {~m}\). The centre of mass of the lamina is at \(O\).

1. Find the value of \(h\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_680_627_1466_797} The lamina is suspended from a vertical string attached to a point \(X\) on the side \(A D\) of the triangle (see diagram). Given the lamina is in equilibrium with \(A D\) horizontal, calculate \(X D\).

1. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
   [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]

\begin{figure}[h] \end{figure} A uniform solid hemisphere of radius \(r\) is joined to a uniform solid right circular cone made of the same material to form a toy. The cone has base radius \(r\) and height \(k r\). The centre of the base of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3. The toy can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
(b) Find the exact value of \(k\)

3 The equation of a curve is \(y = \lambda x ^ { 2 }\), where \(\lambda > 0\). The region bounded by the curve, the \(x\)-axis and the line \(x = a\), where \(a > 0\), is denoted by \(R\). The \(y\)-coordinate of the centroid of \(R\) is \(a\). Show that \(\lambda = \frac { 10 } { 3 a }\).