6.04d Integration: for centre of mass of laminas/solids

4 In this question you may use the following facts: as illustrated in Fig. 4.1, the centre of mass, G, of a uniform thin open hemispherical shell is at the mid-point of OA on its axis of symmetry; the surface area of this shell is \(2 \pi r ^ { 2 }\), where \(r\) is the distance OA. \begin{figure}[h] \end{figure} A perspective view and a cross-section of a dog bowl are shown in Fig. 4.2. The bowl is made throughout from thin uniform material. An open hemispherical shell of radius 8 cm is fitted inside an open circular cylinder of radius 8 cm so that they have a common axis of symmetry and the rim of the hemisphere is at one end of the cylinder. The height of the cylinder is \(k \mathrm {~cm}\). The point O is on the axis of symmetry and at the end of the cylinder. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the centre of mass of the bowl is a distance \(\frac { 64 + k ^ { 2 } } { 16 + 2 k } \mathrm {~cm}\) from O . A version of the bowl for the 'senior dog' has \(k = 12\) and an end to the cylinder, as shown in Fig. 4.3. The end is made from the same material as the original bowl.
2. Show that the centre of mass of this bowl is a distance \(6 \frac { 1 } { 3 } \mathrm {~cm}\) from O . This bowl is placed on a rough slope inclined at \(\theta\) to the horizontal.
3. Assume that the bowl is prevented from sliding and is on the point of toppling. Draw a diagram indicating the position of the centre of mass of the bowl with relevant lengths marked. Calculate the value of \(\theta\).
4. If the bowl is not prevented from sliding, determine whether it will slide when placed on the slope when there is a coefficient of friction between the bowl and the slope of 1.5.

3

1. You are given that the position of the centre of mass, G , of a right-angled triangle cut from thin uniform material in the position shown in Fig. 3.1 is at the point \(\left( \frac { 1 } { 3 } a , \frac { 1 } { 3 } b \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_328_382_360_845} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} A plane thin uniform sheet of metal is in the shape OABCDEFHIJO shown in Fig. 3.2. BDEA and CDIJ are rectangles and FEH is a right angle. The lengths of the sides are shown with each unit representing 1 cm . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_862_906_1032_584} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
   1. Calculate the coordinates of the centre of mass of the metal sheet, referred to the axes shown in Fig. 3.2. The metal sheet is freely suspended from corner B and hangs in equilibrium.
   2. Calculate the angle between BD and the vertical.
2. Part of a framework of light rigid rods freely pin-jointed at their ends is shown in Fig. 3.3. The framework is in equilibrium. All the rods meeting at the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C are shown. The rods connected to \(\mathrm { A } , \mathrm { B }\) and C are connected to the rest of the framework at \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\) and T . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-5_499_734_493_662} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure} There is a tension of 18 N in rod AP and a thrust (compression) of 5 N in rod AQ.
   1. Show the forces internal to the rods acting on the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C .
   2. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CA , stating whether each rod is in tension or compression. [You may leave your answers in surd form. Your working in this part should be consistent with your diagram in part (i).] \(4 P\) and \(Q\) are circular discs of mass 3 kg and 10 kg respectively which slide on a smooth horizontal surface. The discs have the same diameter and move in the line joining their centres with no resistive forces acting on them. The surface has vertical walls which are perpendicular to the line of centres of the discs. This information is shown in Fig. 4 together with the direction you should take as being positive. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-6_430_1404_443_328} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
      1. For what time must a force of 26 N act on P to accelerate it from rest to \(13 \mathrm {~ms} ^ { - 1 }\) ? P is travelling at \(13 \mathrm {~ms} ^ { - 1 }\) when it collides with Q , which is at rest. The coefficient of restitution in this collision is \(e\).
      2. Show that, after the collision, the velocity of P is \(( 3 - 10 e ) \mathrm { ms } ^ { - 1 }\) and find an expression in terms of \(e\) for the velocity of Q.
      3. For what set of values of \(e\) does the collision cause P to reverse its direction of motion?
      4. Determine the set of values of \(e\) for which P has a greater speed than Q immediately after the collision. You are now given that \(e = \frac { 1 } { 2 }\). After P and Q collide with one another, each has a perfectly elastic collision with a wall. P and Q then collide with one another again and in this second collision they stick together (coalesce).
      5. Determine the common velocity of P and Q .
      6. Determine the impulse of Q on P in this collision.

6. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.

1. Write down the distance of the centre of mass from \(A D\).
2. Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\). Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
3. Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
   (4 marks)
   Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical,
4. find the value of \(k\).

4 The region between the curve \(y = 4 - x ^ { 2 }\) and the \(x\)-axis, from \(x = 0\) to \(x = 2\), is occupied by a uniform lamina. The units of the axes are metres.

1. Show that the coordinates of the centre of mass of this lamina are \(( 0.75,1.6 )\). This lamina and another exactly like it are attached to a uniform rod PQ , of mass 12 kg and length 8 m , to form a rigid body as shown in Fig. 4. Each lamina has mass 6.5 kg . The ends of the rod are at \(\mathrm { P } ( - 4,0 )\) and \(\mathrm { Q } ( 4,0 )\). The rigid body lies entirely in the \(( x , y )\) plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
2. Find the coordinates of the centre of mass of the rigid body. The rigid body is freely suspended from the point \(\mathrm { A } ( 2,4 )\) and hangs in equilibrium.
3. Find the angle that PQ makes with the horizontal.

4 In this question, \(a\) is a constant with \(a > 1\).
Fig. 4 shows the region bounded by the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant a\), the \(x\)-axis, and the lines \(x = 1\) and \(x = a\). \begin{figure}[h] \end{figure} This region is occupied by a uniform lamina ABCD , where A is \(( 1,1 ) , \mathrm { B }\) is \(( 1,0 ) , \mathrm { C }\) is \(( a , 0 )\) and D is \(\left( a , \frac { 1 } { a ^ { 2 } } \right)\). The centre of mass of this lamina is \(( \bar { x } , \bar { y } )\).

1. Find \(\bar { x }\) in terms of \(a\), and show that \(\bar { y } = \frac { a ^ { 3 } - 1 } { 6 \left( a ^ { 3 } - a ^ { 2 } \right) }\).
2. In the case \(a = 2\), the lamina is freely suspended from the point A , and hangs in equilibrium. Find the angle which AB makes with the vertical. The region shown in Fig. 4 is now rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
3. Find the \(x\)-coordinate of the centre of mass of this solid of revolution, in terms of \(a\), and show that it is less than 1.5.

4 Fig. 4.1 shows the region \(R\) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 8\). \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of the centre of mass of a uniform solid of revolution obtained by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
2. Find the coordinates of the centre of mass of a uniform lamina in the shape of the region \(R\).
3. Using your answer to part (ii), or otherwise, find the coordinates of the centre of mass of a uniform lamina in the shape of the region (shown shaded in Fig. 4.2) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the line \(y = \frac { 1 } { 2 }\) and the line \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_595_1015_1610_607} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}

4

1. The region bounded by the \(x\)-axis and the semicircle \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(- a \leqslant x \leqslant a\) is occupied by a uniform lamina with area \(\frac { 1 } { 2 } \pi a ^ { 2 }\). Show by integration that the \(y\)-coordinate of the centre of mass of this lamina is \(\frac { 4 a } { 3 \pi }\).
2. A uniform solid cone is formed by rotating the region between the \(x\)-axis and the line \(y = m x\), for \(0 \leqslant x \leqslant h\), through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that the \(x\)-coordinate of the centre of mass of this cone is \(\frac { 3 } { 4 } h\).
      [0pt] [You may use the formula \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
      From such a uniform solid cone with radius 0.7 m and height 2.4 m , a cone of material is removed. The cone removed has radius 0.4 m and height 1.1 m ; the centre of its base coincides with the centre of the base of the original cone, and its axis of symmetry is also the axis of symmetry of the original cone. Fig. 4 shows the resulting object; the vertex of the original cone is V, and A is a point on the circumference of its base. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-5_716_1228_1027_497} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Find the distance of the centre of mass of this object from V . This object is suspended by a string attached to a point Q on the line VA, and hangs in equilibrium with VA horizontal.
   3. Find the distance VQ.

2

1. A uniform solid hemisphere of volume \(\frac { 2 } { 3 } \pi a ^ { 3 }\) is formed by rotating the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), through \(2 \pi\) radians about the \(x\)-axis. Show that the \(x\)-coordinate of the centre of mass of the hemisphere is \(\frac { 3 } { 8 } a\).
2. A uniform lamina is bounded by the \(x\)-axis, the line \(x = 1\), and the curve \(y = 2 - \sqrt { x }\) for \(1 \leqslant x \leqslant 4\). Its corners are \(\mathrm { A } ( 1,1 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 4,0 )\).
   1. Find the coordinates of the centre of mass of the lamina. The lamina is suspended with AB vertical and BC horizontal by light vertical strings attached to A and C , as shown in Fig. 2. The weight of the lamina is \(W\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-2_346_684_1672_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure}
   2. Find the tensions in the two strings in terms of \(W\).

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_705_501_260_863} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The region \(R\), shown in Fig. 4.1, is bounded by the curve \(x ^ { 2 } - y ^ { 2 } = k ^ { 2 }\) for \(k \leqslant x \leqslant 4 k\) and the line \(x = 4 k\), where \(k\) is a positive constant. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when \(R\) is rotated about the \(x\)-axis.
2. A uniform lamina occupies the region bounded by the curve \(y = \frac { x ^ { 3 } } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant 2 a\), the \(x\)-axis and the line \(x = 2 a\), where \(a\) is a positive constant. The vertices of the lamina are \(\mathrm { O } ( 0,0 ) , \mathrm { A } ( 2 a , 8 a )\) and \(\mathrm { B } ( 2 a , 0 )\), as shown in Fig. 4.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_714_509_1546_858} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Find the coordinates of the centre of mass of the lamina.
   2. The lamina is freely suspended from the point A and hangs in equilibrium. Find the angle that AB makes with the vertical.

4

1. The region \(T\) is bounded by the \(x\)-axis, the line \(y = k x\) for \(a \leqslant x \leqslant 3 a\), the line \(x = a\) and the line \(x = 3 a\), where \(k\) and \(a\) are positive constants. A uniform frustum of a cone is formed by rotating \(T\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this frustum.
2. A uniform lamina occupies the region (shown in Fig. 4) bounded by the \(x\)-axis, the curve \(y = 16 \left( 1 - x ^ { - \frac { 1 } { 3 } } \right)\) for \(1 \leqslant x \leqslant 8\) and the line \(x = 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-4_368_519_1439_772} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the coordinates of the centre of mass of this lamina. A hole is made in the lamina by cutting out a circular disc of area 5 square units. This causes the centre of mass of the lamina to move to the point \(( 5,3 )\).
   2. Find the coordinates of the centre of the hole.

4

1. The region enclosed between the curve \(y = x ^ { 4 }\) and the line \(y = h\) (where \(h\) is positive) is rotated about the \(y\)-axis to form a uniform solid of revolution. Find the \(y\)-coordinate of the centre of mass of this solid.
2. The region \(A\) is bounded by the \(x\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(x = 4\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(y = 6\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-3_572_513_1779_778} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. A uniform lamina occupies the region \(A\). Show that the \(x\)-coordinate of the centre of mass of this lamina is 2.56 , and find the \(y\)-coordinate.
   2. Using your answer to part (i), or otherwise, find the coordinates of the centre of mass of a uniform lamina occupying the region \(B\).

4 The region bounded by the curve \(y = \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.

1. Find the \(x\)-coordinate of the centre of mass of this solid. From this solid, the cylinder with radius 1 and length 3 with its axis along the \(x\)-axis (from \(x = 1\) to \(x = 4\) ) is removed.
2. Show that the centre of mass of the remaining object, Q , has \(x\)-coordinate 3 . This object Q has weight 96 N and it is supported, with its axis of symmetry horizontal, by a string passing through the cylindrical hole and attached to fixed points A and B (see Fig. 4). AB is horizontal and the sections of the string attached to A and B are vertical. There is sufficient friction to prevent slipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-5_837_819_1034_628} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
3. Find the support forces, \(R\) and \(S\), acting on the string at A and B
   (A) when the string is light,
   (B) when the string is heavy and uniform with a total weight of 6 N .

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, the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 3\), and the line \(x = \ln 3\). Find, in an exact form, the coordinates of the centre of mass of this lamina.
2. A region is bounded by the \(x\)-axis, the curve \(y = \frac { 6 } { x ^ { 2 } }\) for \(2 \leqslant x \leqslant a\) (where \(a > 2\) ), the line \(x = 2\) and the line \(x = a\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
   1. Show that the \(x\)-coordinate of the centre of mass of this solid is \(\frac { 3 \left( a ^ { 3 } - 4 a \right) } { a ^ { 3 } - 8 }\).
   2. Show that, however large the value of \(a\), the centre of mass of this solid is less than 3 units from the origin.

3 In this question, give your answers in an exact form.
The region \(R _ { 1 }\) (shown in Fig. 3) is bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 5\), and the curve \(y = \frac { 1 } { x }\) for \(1 \leqslant x \leqslant 5\).

1. A uniform solid of revolution is formed by rotating the region \(R _ { 1 }\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
2. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-4_849_841_735_651} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The region \(R _ { 2 }\) is bounded by the \(y\)-axis, the lines \(y = 1\) and \(y = 5\), and the curve \(y = \frac { 1 } { x }\) for \(\frac { 1 } { 5 } \leqslant x \leqslant 1\). The region \(R _ { 3 }\) is the square with vertices \(( 0,0 ) , ( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\).
3. Write down the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 2 }\).
4. Find the coordinates of the centre of mass of a uniform lamina occupying the region consisting of \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) (shown shaded in Fig. 3).

4 The region \(A\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, the \(y\)-axis and the line \(x = 3\). The region \(B\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(y\)-axis and the line \(y = 14\). These regions are shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(A\).
2. The region \(B\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(R\). Find the \(y\)-coordinate of the centre of mass of the solid \(R\).
3. The region \(A\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(S\). Using your answer to part (ii), or otherwise, find the \(y\)-coordinate of the centre of mass of the solid \(S\).

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

1. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.

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}

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis and the curve \(y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant a\). Find the coordinates of the centre of mass of this lamina.
2. The region \(A\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(x = 3\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(y = 5\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(A\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   2. Using your answer to part (i), or otherwise, find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. \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 \(\alpha\).

5.

1. Use integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is \(\frac { 3 } { 4 } h\) from the vertex of the cone.
   (6 marks) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius \(r\) so that the remaining solid consists of a hemisphere of radius \(r\) and a cone of height \(r\) and base radius \(r\) as shown in Figure 3.
2. Find the distance of the centre of mass of the paperweight from its vertex.
   (7 marks)
   

6. \begin{figure}[h] \end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.

1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
   (7 marks)
   The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
2. Find tan \(\alpha\).
   (6 marks)

6. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
   (3 marks)