Composite solid with cone and cylinder

5. \begin{figure}[h] \end{figure} Figure 3 shows a uniform solid \(S\) formed by joining the plane faces of two solid right circular cones, of base radius \(r\), so that the centres of their bases coincide at \(O\). One cone, with vertex \(V\), has height \(4 r\) and the other cone has height \(k r\), where \(k > 4\)

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (4) The point \(A\) lies on the circumference of the common base of the cones. The solid is placed on a horizontal surface with VA in contact with the surface. Given that \(S\) rests in equilibrium,
2. find the greatest possible value of \(k\). When \(S\) is suspended from \(A\) and hangs freely in equilibrium, \(O A\) makes an angle of \(12 ^ { \circ }\) with the downward vertical.
3. Find the value of \(k\).

1. \begin{figure}[h] \end{figure} A uniform solid \(S\) consists of a right solid circular cone of base radius \(r\) and a right solid cylinder, also of radius \(r\). The cone has height \(4 h\) and the centre of the plane face of the cone is \(O\). The cylinder has height \(3 h\). The cone and cylinder are joined so that the plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Figure 1. Find the distance from \(O\) to the centre of mass of \(S\).

5. \begin{figure}[h] \end{figure} A toy is formed by joining a uniform solid right circular cone, of base radius \(3 r\) and height \(4 r\), to a uniform solid cylinder, also of radius \(3 r\) and height \(4 r\). The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is \(O\).

1. Find the distance of the centre of mass of the toy from \(O\). The point \(A\) lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from \(A\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical. The toy is placed with the curved surface of the cone on horizontal ground.
3. Determine whether the toy will topple.
   (4)

4. A uniform solid \(S\) consists of two right circular cones of base rate
has height \(2 h\) and the centre of the plane face of this cone is \(O\). T \(k h\) where \(k > 2\). The two cones are joined so that their plane fac
Figure 2 .

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { h } { 4 } ( k - 2 )\)

5. \includegraphics[max width=\textwidth, alt={}, center]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-3_423_357_918_847} A uniform solid, \(S\), is placed with its plane face on horizontal ground. The solid consists of a right circular cylinder, of radius \(r\) and height \(r\), joined to a right circular cone, of radius \(r\) and height \(h\). The plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Fig. 3.

1. Show that the distance of the centre of mass of \(S\) from the ground is $$\frac { 6 r ^ { 2 } + 4 r h + h ^ { 2 } } { 4 ( 3 r + h ) }$$ (8) The solid is now placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is rough enough to prevent \(S\) from sliding. Given that \(h = 2 r\), and that \(S\) is on the point of toppling,
2. find, to the nearest degree, the value of \(\alpha\).
   (5)

\includegraphics{figure_4} A uniform solid circular cone, of vertical height \(4r\) and radius \(2r\), is attached to a uniform solid cylinder, of height \(3r\) and radius \(kr\), where \(k\) is a constant less than 2. The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram). The cone and the cylinder are made of the same material.

1. Show that the distance of the centre of mass of the combined solid from the vertex of the cone is $$\frac{(99k^2 + 96)r}{18k^2 + 32}.$$ [4]

\includegraphics{figure_2} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre \(O\) of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is \(r\) and the radius of the base of the cone is \(2r\). The height of the cone and the height of the cylinder are each \(h\). The centre of mass of the model is at the point \(G\).

1. Show that \(OG = \frac{1}{14}h\). [8]

The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{r}{7h}\). The desk top is rough enough to prevent slipping and the model is about to topple.

1. Find \(r\) in terms of \(h\). [4]

\includegraphics{figure_3} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac{1}{4}(x - 2)^2\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid \(S\) is made by rotating \(R\) through \(360°\) about the \(x\)-axis. Using integration,

1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\), [4]
2. show that the centre of mass of \(S\) is \(\frac{4}{5}\) cm from its plane face. [7]

\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm. One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10W\) newtons and the weight of \(S\) is \(2W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.

1. Find the magnitude of the force of the support \(A\) on the tool. [5]