Solid on inclined plane - toppling

3. \begin{figure}[h] \end{figure} The shaded region in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 2\) and the curve with equation \(y = \frac { 1 } { 4 } x ( 3 - x )\).
This region is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a solid of revolution which is used to model a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 5 } \pi\)

1. Use the model and algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 31 } { 24 }\) The solid \(S\) is placed with its circular face on a rough plane which is inclined at \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. The solid \(S\) is on the point of toppling.
2. Find the value of \(\alpha\)

7. Diagram NOT accurately drawn \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 } x ( 6 - x )\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 1. The unit of length on both axes is 1 cm . A uniform solid \(P\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(P\) is, to 3 significant figures, 1.42 cm from its plane face. The uniform solid \(P\) is placed with its plane face on an inclined plane which makes an angle \(\theta\) with the horizontal. Given that the plane is sufficiently rough to prevent \(P\) from sliding and that \(P\) is on the point of toppling when \(\theta = \alpha\),
2. find the angle \(\alpha\). Given instead that \(P\) is on the point of sliding down the plane when \(\theta = \beta\) and that the coefficient of friction between \(P\) and the plane is 0.3 ,
3. find the angle \(\beta\).

4

1. The region bounded by the curve \(y = x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 2\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8]
2. The region bounded by the circular arc \(y = \sqrt { 4 - x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 1\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_627_499_593_785} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Show that the \(x\)-coordinate of the centre of mass of this solid of revolution is 1.35 . This solid is placed on a rough horizontal surface, with its flat face in a vertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig. 4.2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_573_613_1662_728} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the least possible coefficient of friction between the solid and the horizontal surface.

4

1. A uniform solid of revolution is obtained by rotating through \(2 \pi\) radians about the \(y\)-axis the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the \(y\)-axis.
   1. Find the \(y\)-coordinate of the centre of mass of this solid. The solid is now placed on a rough plane inclined at an angle \(\theta\) to the horizontal. It rests in equilibrium with its circular face in contact with the plane as shown in Fig. 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-4_511_568_616_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Given that the solid is on the point of toppling, find \(\theta\).
2. Find the \(y\)-coordinate of the centre of mass of a uniform lamina in the shape of the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(- 2 \leqslant x \leqslant 2\), and the \(x\)-axis.

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = 3 - \sqrt { x }\) for \(0 \leqslant x \leqslant 9\). Find the coordinates of the centre of mass of this lamina.
2. Fig. 4.1 shows the region bounded by the line \(x = 2\) and the part of the circle \(y ^ { 2 } = 25 - x ^ { 2 }\) for which \(2 \leqslant x \leqslant 5\). This region is rotated about the \(x\)-axis to form a uniform solid of revolution \(S\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of \(S\). The solid \(S\) rests in equilibrium with its curved surface in contact with a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of \(S\). This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle \(\theta\) that AB makes with the horizontal.

4 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { - x }\) and the line \(x = k\), where \(k\) is a positive constant.

1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
2. The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
   [diagram]

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Show that, whatever the value of \(k\), the solid will not topple.
3. A uniform lamina occupies the region \(R\). Find, in terms of \(k\), the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}

5. \begin{figure}[h] \end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height \(2 r\) and base radius \(r\), to one of the plane surfaces of a cylinder of height \(h\) and base radius \(r\) as shown in Figure 2. Using this model,

1. show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle \(\alpha\) to the horizontal. Given that the firework does not slip or topple and that \(h = 4 r\),
2. Find, correct to the nearest degree, the maximum value of \(\alpha\).

6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6

1. Find the centre of mass of this solid.
   6
2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)

4. \begin{figure}[h] \end{figure} A uniform solid cylinder of base radius \(r\) and height \(\frac { 4 } { 3 } r\) has the same density as a uniform solid hemisphere of radius \(r\). The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid \(S\) shown in Figure 3. The point \(O\) is the centre of the plane face of \(S\).

1. Show that the distance from \(O\) to the centre of mass of \(S\) is \(\frac { 73 } { 72 } r\) The solid \(S\) is placed with its plane face on a rough horizontal plane. The coefficient of friction between \(S\) and the plane is \(\mu\). A horizontal force \(P\) is applied to the highest point of \(S\). The magnitude of \(P\) is gradually increased.
2. Find the range of values of \(\mu\) for which \(S\) will slide before it starts to tilt.