6.04c Composite bodies: centre of mass

4. \begin{figure}[h] \end{figure} Figure 2 shows a lamina \(L\). It is formed by removing a square \(P Q R S\) from a uniform triangle \(A B C\). The triangle \(A B C\) is isosceles with \(A C = B C\) and \(A B = 12 \mathrm {~cm}\). The midpoint of \(A B\) is \(D\) and \(D C = 8 \mathrm {~cm}\). The vertices \(P\) and \(Q\) of the square lie on \(A B\) and \(P Q = 4 \mathrm {~cm}\). The centre of the square is \(O\). The centre of mass of \(L\) is at \(G\).

1. Find the distance of \(G\) from \(A B\). When \(L\) is freely suspended from \(A\) and hangs in equilibrium, the line \(A B\) is inclined at \(25 ^ { \circ }\) to the vertical.
2. Find the distance of \(O\) from \(D C\).

1. Three particles of mass \(3 m , 2 m\) and \(k m\) are placed at the points whose coordinates are \(( 1,5 ) , ( 6,4 )\) and \(( a , 1 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( 3,3 )\).

Find

1. the value of \(k\),
2. the value of \(a\).

3. [The centre of mass of a semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h] \end{figure} Figure 2 shows the uniform lamina \(A B C D E\), such that \(A B D E\) is a square with sides of length \(2 a\) and \(B C D\) is a semicircle with diameter \(B D\).

1. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 20 a } { 3 ( 8 + \pi ) }\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
2. Find, to the nearest degree, the angle that \(D E\) makes with the downward vertical.

3. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles triangle \(A B C\), where \(A B = A C\). The mid-point of \(B C\) is \(M , A M = 30 \mathrm {~cm}\) and \(B M = 40 \mathrm {~cm}\). The mid-points of \(A C\) and \(A B\) are \(D\) and \(E\) respectively. The triangular portion \(A D E\) is removed leaving a uniform plane lamina \(B C D E\) as shown in Fig. 2.

1. Show that the centre of mass of the lamina \(B C D E\) is \(6 \frac { 2 } { 3 } \mathrm {~cm}\) from \(B C\).
   (6 marks)
   The lamina \(B C D E\) is freely suspended from \(D\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle which \(D E\) makes with the vertical.
   (3 marks)

7. \includegraphics[max width=\textwidth, alt={}, center]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-4_360_472_1105_815} A uniform plane lamina \(A B C D E\) is formed by joining a uniform square \(A B D E\) with a uniform triangular lamina \(B C D\), of the same material, along the side \(B D\), as shown in Fig. 2. The lengths \(A B , B C\) and \(C D\) are \(18 \mathrm {~cm} , 15 \mathrm {~cm}\) and 15 cm respectively.

1. Find the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle which \(B D\) makes with the vertical.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the region \(R\) bounded by the curve with equation \(y = \mathrm { e } ^ { - x }\), the line \(x = 1\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac { \pi } { 2 } \left( 1 - \mathrm { e } ^ { - 2 } \right)\).
2. Find, in terms of e, the distance of the centre of mass of \(S\) from \(O\).

5. A solid \(S\) consists of a uniform solid hemisphere of radius \(r\) and a uniform solid circular cylinder of radius \(r\) and height \(3 r\). The circular face of the hemisphere is joined to one of the circular faces of the cylinder, so that the centres of the two faces coincide. The other circular face of the cylinder has centre \(O\). The mass per unit volume of the hemisphere is \(3 k\) and the mass per unit volume of the cylinder is \(k\).

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 9 r } { 4 }\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-07_501_1082_653_422} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The solid \(S\) is held in equilibrium by a horizontal force of magnitude \(P\). The circular face of \(S\) has one point in contact with a fixed rough horizontal plane and is inclined at an angle \(\alpha\) to the horizontal. The force acts through the highest point of the circular face of \(S\) and in the vertical plane through the axis of the cylinder, as shown in Figure 2. The coefficient of friction between \(S\) and the plane is \(\mu\). Given that \(S\) is on the point of slipping along the plane in the same direction as \(P\),
2. show that \(\mu = \frac { 1 } { 8 } ( 9 - 4 \cot \alpha )\).

2. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\) which is bounded by the curve with equation \(y = \frac { 3 } { x ^ { 2 } }\), the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis, as shown in Figure 1. The centre of mass of the lamina has coordinates \(( \bar { x } , \bar { y } )\).
Use algebraic integration to find

1. the value of \(\bar { x }\),
2. the value of \(\bar { y }\).

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 \(k\).

3. \begin{figure}[h] \end{figure} A uniform right circular solid cylinder has radius \(4 a\) and height \(6 a\). A solid hemisphere of radius \(3 a\) is removed from the cylinder forming a solid \(S\). The upper plane face of the cylinder coincides with the plane face of the hemisphere. The centre of the upper plane face of the cylinder is \(O\) and this is also the centre of the plane face of the hemisphere, as shown in Figure 2. Find the distance from \(O\) to the centre of mass of \(S\).
(6)

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} The region \(R\), shown shaded in Figure 3, is bounded by the circle with centre \(O\) and radius \(r\), the line with equation \(x = \frac { 3 } { 5 } r\) and the \(x\)-axis. The region is rotated through one complete revolution about the \(x\)-axis to form a uniform solid \(S\).

1. Use algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 48 } { 65 } r\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre \(A\) and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is \(5 M \mathrm {~kg}\) and the mass of the liquid is \(2 M \mathrm {~kg}\).
2. Find, to 3 significant figures, the distance from \(A\) to the centre of mass of the bowl with its liquid.

4. \begin{figure}[h] \end{figure} A uniform right solid cone \(C\) has diameter \(6 a\) and height \(8 a\), as shown in Figure 3.
The solid \(S\) is formed by removing a cone of height \(4 a\) from the top of \(C\) and then removing an identical, inverted cone. The vertex of the removed cone is at the point \(O\) in the centre of the base of \(C\), as shown in Figure 4. \begin{figure}[h] \end{figure}

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (5) The point \(A\) lies on the circumference of the base of \(S\) and the point \(B\) lies on the circumference of the top of \(S\). The points \(O\), \(A\) and \(B\) all lie in the same vertical plane, as shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_248_449_1845_749} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The solid \(S\) is freely suspended from the point \(B\) and hangs in equilibrium.
2. Find the size of the angle that \(A B\) makes with the downward vertical.

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(2 a\) and height \(k a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(2 a\), is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Show that the distance from \(O\) to the centre of mass of \(C\) is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(k\).

3. A square ABCD of side 4a is made from thin uniform cardboard. The centre of the square is 0 . A circle with centre 0 and radius \(\frac { 7 a } { 4 }\) is then removed from the square to form a template T, shown shaded in Figure 3.
A right conical shell, with no base, has radius \(\frac { 7 a } { 4 }\) and perpendicular height \(6 a\).
The shell is made of the same thin uniform cardboard as T.
The shell is attached to T so that the circumference of the end of the shell coincides with the circumference of the circle centre 0 , to form the hat H , shown in Figure 4.
[0pt] [The surface area of a right conical shell of radius r and slant height I is \(\pi r l\).]

1. Show that the exact distance of the centre of mass of H from O is $$\frac { 175 \pi a } { ( 63 \pi + 128 ) }$$ A fixed rough plane is inclined to the horizontal at an angle \(\alpha\). The hat H is placed on the plane, with ABCD in contact with the plane, and AB parallel to a line of greatest slope of the plane. The plane is sufficiently rough to prevent the hat from sliding down the plane. Given that the hat is on the point of toppling,
2. find the exact value of \(\tan \alpha\), giving your answer in simplest form.

7. \begin{figure}[h] \end{figure} Diagram not drawn to scale A uniform right circular solid cylinder has radius \(3 a\) and height \(2 a\). A right circular cone of height \(\frac { 3 a } { 2 }\) and base radius \(2 a\) is removed from the cylinder to form a solid \(S\), as shown in Figure 4. The plane face of the cone coincides with the upper plane face of the cylinder and the centre \(O\) of the plane face of the cone is also the centre of the upper plane face of the cylinder.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 69 a } { 64 }\). The point \(A\) is on the open face of \(S\) such that \(O A = 3 a\), as shown in Figure 4. The solid is now suspended from \(A\) and hangs freely in equilibrium.
2. Find the angle between \(O A\) and the horizontal.
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_543_826_1653_557} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The solid is now placed on a rough inclined plane with the face through \(A\) in contact with the inclined plane, as shown in Figure 5. The solid rests in equilibrium on this plane. The coefficient of friction between the plane and \(S\) is 0.6 and the plane is inclined at an angle \(\phi ^ { \circ }\) to the horizontal. Given that \(S\) is on the point of sliding down the plane,
3. show that \(\phi = 31\) to 2 significant figures.

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

2. \begin{figure}[h] \end{figure} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac { 1 } { 3 }\). The cone is freely suspended by a string attached at a point \(A\) on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
   \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
2. Find the distance of the centre of mass of \(T\) from its plane face.

2. \begin{figure}[h] \end{figure} A light inextensible string of length \(8 l\) has its ends fixed to two points \(A\) and \(B\), where \(A\) is vertically above \(B\). A small smooth ring of mass \(m\) is threaded on the string. The ring is moving with constant speed in a horizontal circle with centre \(B\) and radius 3l, as shown in Fig. 2. Find

1. the tension in the string,
2. the speed of the ring.
3. State briefly in what way your solution might no longer be valid if the ring were firmly attached to the string.
   (1) \section*{3.} \section*{Figure 3}
   \includegraphics[max width=\textwidth, alt={}]{044c5866-0a12-4309-8ced-b463e1615fb0-3_564_1051_438_541}
   A child's toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig. 3. The cylinder and the hemisphere each have radius \(r\), and the height of the cylinder is \(h\). The material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the cylinder vertical.

6. Figure 2 \includegraphics[max width=\textwidth, alt={}, center]{c4b453e7-8a32-458b-8041-58c9e4ef9533-5_691_1067_241_584} A uniform solid cylinder has radius \(2 a\) and height \(\frac { 3 } { 2 } a\). A hemisphere of radius \(a\) is removed from the cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder, and the centre \(O\) of the hemisphere is also the centre of this plane face, as shown in Fig. 2. The remaining solid is \(S\).

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (6) The lower plane face of \(S\) rests in equilibrium on a desk lid which is inclined at an angle \(\theta\) to the horizontal. Assuming that the lid is sufficiently rough to prevent \(S\) from slipping, and that \(S\) is on the point of toppling when \(\theta = \alpha\),
2. find the value of \(\alpha\).
   (3) Given instead that the coefficient of friction between \(S\) and the lid is 0.8 , and that \(S\) is on the point of sliding down the lid when \(\theta = \beta\),
3. find the value of \(\beta\).
   (3)

4. \begin{figure}[h] \end{figure} A body consists of a uniform solid circular cylinder \(C\), together with a uniform solid hemisphere \(H\) which is attached to \(C\). The plane face of \(H\) coincides with the upper plane face of \(C\), as shown in Figure 2. The cylinder \(C\) has base radius \(r\), height \(h\) and mass 3M. The mass of \(H\) is \(2 M\). The point \(O\) is the centre of the base of \(C\).

1. Show that the distance of the centre of mass of the body from \(O\) is $$\frac { 14 h + 3 r } { 20 } .$$ The body is placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 4 } { 3 }\). The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
2. find \(h\) in terms of \(r\).

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_616_431_1420_778}
   \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
2. Find the distance of the centre of mass of \(T\) from its plane face.

3. \begin{figure}[h] \end{figure} A uniform solid \(S\) is formed by taking a uniform solid right circular cone, of base radius \(2 r\) and height \(2 h\), and removing the cone, with base radius \(r\) and height \(h\), which has the same vertex as the original cone, as shown in Figure 1.

1. Show that the distance of the centre of mass of \(S\) from its larger plane face is \(\frac { 11 } { 28 } h\). The solid \(S\) lies with its larger plane face on a rough table which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The table is sufficiently rough to prevent \(S\) from slipping. Given that \(h = 2 r\),
2. find the greatest value of \(\theta\) for which \(S\) does not topple.