Centre of Mass 1

A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]

3 Three particles are attached to a light rod, \(A B\), of length 40 cm The particles are attached at \(A , B\) and the midpoint of the rod.
The particle at \(A\) has mass 5 kg
The particle at \(B\) has mass 1 kg
The particle at the midpoint has mass 4 kg
Find the distance of the centre of mass of this system from the midpoint of the rod.
Circle your answer.
[0pt] [1 mark] \(4 \mathrm {~cm} \quad 8 \mathrm {~cm} \quad 12 \mathrm {~cm} \quad 28 \mathrm {~cm}\) Turn over for the next question

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{3}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]
2. The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\). Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

3 A uniform lamina \(A B C D E\) consists of a square \(A C D E\) and an equilateral triangle \(A B C\) which are joined along their common edge \(A C\) to form a pentagon whose sides are each 8 cm in length.

1. Calculate the distance of the centre of mass of the lamina from \(A C\).
2. The lamina is freely suspended from \(A\) and hangs in equilibrium. Calculate the angle that \(A C\) makes with the vertical.

1. \hspace{0pt} [The centre of mass of a semicircular arc of radius \(r\) is \(\frac { 2 r } { \pi }\) from the centre.]

\begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework,

• \(A B C\) is straight and has length \(25 a\)
• \(A D E\) is straight and has length \(24 a\)
• \(A B D\) is a semicircular arc of radius \(7 a\)
• \(E C = 7 a\)
• angle \(A E C = 90 ^ { \circ }\)
• the points \(A , B , C , D\) and \(E\) all lie in the same plane

The distance of the centre of mass of the framework from \(A E\) is \(d\).

1. Show that \(d = \frac { 53 } { 2 ( 7 + \pi ) } a\) The framework is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at angle \(\alpha ^ { \circ }\) to the downward vertical.
2. Find the value of \(\alpha\).

4. The diagram shows a uniform lamina consisting of a rectangular section \(G P Q E\) with a semi-circular section EFG of radius 4 cm . Quadrants \(A P B\) and \(C Q D\) each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

1. Write down the distance of the centre of mass of the lamina \(A B C D E F G\) from \(A G\).
2. Determine the distance of the centre of mass of the lamina \(A B C D E F G\) from \(B C\).
3. The lamina \(A B C D E F G\) is suspended freely from the point \(E\) and hangs in equilibrium. Calculate the angle EG makes with the vertical.

5. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F\), shown in Figure 2, consists of two identical rectangles with sides of length \(a\) and \(3 a\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(E\). The lamina, with the attached particle, is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at an angle \(\theta\) to the downward vertical. Given that \(\tan \theta = \frac { 4 } { 7 }\), find the value of \(k\).
(10)

4. \begin{figure}[h] \end{figure} The uniform plane lamina \(A B C D E F\) shown in Figure 1 is made from two identical rhombuses. Each rhombus has sides of length \(a\) and angle \(B A D =\) angle \(D A F = \theta\). The centre of mass of the lamina is \(0.9 a\) from \(A\).

1. Show that \(\cos \theta = 0.8\) The weight of the lamina is \(W\). A particle of weight \(k W\) is fixed to the lamina at the point \(A\). The lamina is freely suspended from \(B\) and hangs in equilibrium with \(D A\) horizontal.
2. Find the value of \(k\).

3. The diagram below shows a lamina \(A B C D E\) which is made of a uniform material. It consists of a rectangle \(A B D E\) with \(A B = 6 a\) and \(A E = 8 a\), together with an isosceles triangle \(B C D\) with \(B C = D C = 5 a\). A semicircle, with its centre at the midpoint of \(A E\) and radius \(3 a\), is removed from \(A B D E\). \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-3_606_703_603_680}

1. Write down the distance of the centre of mass of the lamina \(A B C D E\) from \(A B\).
2. Show that the distance of the centre of mass of the lamina \(A B C D E\) from \(A E\) is \(\frac { 140 } { 40 - 3 \pi } a\).
3. The lamina \(A B C D E\) is freely suspended from the point \(D\) and hangs in equilibrium.
   1. Calculate the angle that \(B D\) makes with the vertical.
   2. The mass of the lamina is \(M\). When a particle of mass \(k M\) is attached at the point \(C\), the lamina hangs in equilibrium with \(A B\) horizontal. Determine the value of \(k\).

\includegraphics{figure_7}

3 \begin{figure}[h] \end{figure} A uniform solid cylinder has mass 8 kg and height 16 cm . A uniform solid cone, whose base radius is the same as the radius of the cylinder, has mass 2 kg and height 12 cm . A composite solid is formed by joining the cylinder and cone so that the base of the cone coincides with one end of the cylinder (see Fig. 1).

1. Show that the centre of mass of the composite solid is 10.2 cm from its base. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_401_444_1877_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The composite solid is held with a point on the circumference of its base in contact with a horizontal table. The base makes an angle \(\theta ^ { \circ }\) with the table (see Fig. 2, which shows a cross-section). When the cone is released it moves towards the equilibrium position in which its base is in contact with the table.
2. Given that the radius of the base is 4 cm , find the greatest possible value of \(\theta\), correct to 1 decimal place.

3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

\includegraphics{figure_1}

4 The diagram shows a uniform lamina which is in the shape of two identical rectangles \(A X G H\) and \(Y B C D\) and a square \(X Y E F\), arranged as shown. The length of \(A X\) is 10 cm , the length of \(X Y\) is 10 cm and the length of \(A H\) is 30 cm . \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-3_1183_1278_513_374}

1. Explain why the centre of mass of the lamina is 15 cm from \(A H\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from the point \(H\). Find, to the nearest degree, the angle between \(H G\) and the horizontal when the lamina is in equilibrium.

3 Two identical uniform rectangular laminas, P and Q , each having length \(k a\) and width \(a\) are fixed together, in the same plane, to form a lamina R.
With reference to coordinate axes, the corners of P are at ( 0,0 ), ( \(k a , 0\) ), ( \(k a , a\) ) and ( \(0 , a\) ) and the corners of Q are at \(( k a , 0 ) , ( k a + a , 0 ) , ( k a + a , k a )\) and \(( k a , k a )\), as shown in Fig. 3. \begin{figure}[h] \end{figure} Determine the range of values of \(k\) for which the centre of mass of R lies outside the boundary of R.

\includegraphics{figure_1} A uniform lamina \(ABCD\) consists of two isosceles triangles \(ABD\) and \(BCD\). The diagonals of \(ABCD\) meet at the point \(O\). The length of \(AO\) is \(3a\), the length of \(OC\) is \(6a\) and the length of \(BD\) is \(16a\) (see diagram). Find the distance of the centre of mass of the lamina from \(DB\). [3]

3 A uniform circular lamina, of radius 4 cm and mass 0.4 kg , has a centre \(O\), and \(A B\) is a diameter. To create a medal, a smaller uniform circular lamina, of radius 2 cm and mass 0.1 kg , is attached so that the centre of the smaller lamina is at the point \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_671_878_513_598}

1. Explain why the centre of mass of the medal is on the line \(A B\).
2. Find the distance of the centre of mass of the medal from the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_1259_1705_1448_155}
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-07_2484_1709_223_153}

\includegraphics{figure_4} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. AB = 5a, AE = 4a and BC = CD. The point F is the midpoint of BD and FC = a.

1. Find, in terms of \(a\), the exact distance of the centre of mass of the lamina from AE. [4]

The lamina is freely suspended from B and hangs in equilibrium.

1. Find the angle between AB and the downward vertical. [2]

An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).

1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]

The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),

1. find the distance of the centre of mass of the filled container from \(O\). [5]

2. A closed container \(C\) consists of a thin uniform hollow hemispherical bowl of radius \(a\), together with a lid. The lid is a thin uniform circular disc, also of radius \(a\). The centre \(O\) of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.

1. Show that the centre of mass of \(C\) is at a distance \(\frac { 1 } { 3 } a\) from \(O\). The container \(C\) has mass \(M\). A particle of mass \(\frac { 1 } { 2 } M\) is attached to the container at a point \(P\) on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with \(P , O\) and the point of contact in the same vertical plane.
2. Find, to the nearest degree, the angle made by the line \(P O\) with the horizontal.

7 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-4_553_630_258_753} The diagram shows a container which consists of a bowl of weight 14 N and a handle of weight 8 N . The bowl of the container is in the form of a uniform hemispherical shell with centre \(O\) and radius 0.3 m . The handle is in the form of a uniform semicircular arc of radius 0.3 m and is freely hinged to the bowl at \(A\) and \(B\), where \(A B\) is a diameter of the bowl.

1. Calculate the distance of the centre of mass of the container from \(O\) for the position indicated in the diagram, where the handle is perpendicular to the rim of the bowl.
2. Show that the distance of the centre of mass of the container from \(O\) when the handle lies on the rim of the bowl is 0.118 m , correct to 3 significant figures. In the case when the handle lies on the rim of the bowl, the container rests in equilibrium with the curved surface of the bowl on a horizontal table.
3. Find the angle which the plane containing the rim of the bowl makes with the horizontal.

2 \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-2_319_908_438_616} A uniform solid cone has height 20 cm and base radius \(4 \mathrm {~cm} . P Q\) is a diameter of the base of the cone. The cone is held in equilibrium, with \(P\) in contact with a horizontal surface and \(P Q\) vertical, by a force applied at \(Q\). This force has magnitude 3 N and acts parallel to the axis of the cone (see diagram). Calculate the mass of the cone.

3 \(A B C D E F\) is the L -shaped cross-section of a uniform solid. This cross-section passes through the centre of mass of the solid and has dimensions as shown in Fig. 1.

1. Find the distance of the centre of mass of the solid from the edge \(A B\) of the cross-section.
   [diagram]
   The solid rests in equilibrium with the face containing the edge \(A F\) of the cross-section in contact with a horizontal table. The weight of the solid is \(W\) N. A horizontal force of magnitude \(P\) N is applied to the solid at the point \(B\), in the direction of \(B C\) (see Fig. 2). The table is sufficiently rough to prevent sliding.
2. Find \(P\) in terms of \(W\), given that the equilibrium of the solid is about to be broken.

5. \begin{figure}[h] \end{figure} A shop sign is modelled as a uniform rectangular lamina \(A B C D\) with a semicircular lamina removed. The semicircle has radius \(a , B C = 4 a\) and \(C D = 2 a\).
The centre of the semicircle is at the point \(E\) on \(A D\) such that \(A E = d\), as shown in Figure 3.

1. Show that the centre of mass of the sign is \(\frac { 44 a } { 3 ( 16 - \pi ) }\) from \(A D\). The sign is suspended using vertical ropes attached to the sign at \(A\) and at \(B\) and hangs in equilibrium with \(A B\) horizontal. The weight of the sign is \(W\) and the ropes are modelled as light inextensible strings.
2. Find, in terms of \(W\) and \(\pi\), the tension in the rope attached at \(B\). The rope attached at \(B\) breaks and the sign hangs freely in equilibrium suspended from \(A\), with \(A D\) at an angle \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 11 } { 18 }\)
3. find \(d\) in terms of \(a\) and \(\pi\).

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

2 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-04_606_376_260_881} A uniform object is made by joining together three solid cubes with edges \(3 \mathrm {~m} , 2 \mathrm {~m}\) and 1 m . The object has an axis of symmetry, with the cubes stacked vertically and the cube of edge 2 m between the other two cubes (see diagram).

1. Calculate the distance of the centre of mass of the object above the base of the largest cube.
   The smallest cube is now removed from the object. It is replaced by a heavier uniform cube with 1 m edges which is made of a different material. The centre of mass of the object is now at the base of the 2 m cube.
2. Find the ratio of the masses of the two cubes of edge 1 m .

6 A uniform solid consists of a hemisphere with centre \(O\) and radius 0.6 m joined to a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere coincides with one of the plane faces of the cylinder.

1. Calculate the distance of the centre of mass of the solid from \(O\).
   [0pt] [The volume of a hemisphere of radius \(r\) is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
2. \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-4_547_631_593_797} A cylindrical hole, of length 0.48 m , starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at \(O\). Show that the area of the cross-section of the hole is \(\frac { 3 } { 16 } \pi \mathrm {~m} ^ { 2 }\).
3. It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at \(O\). State the increase in the length of the hole.

2. You are given that the centre of mass of a uniform solid cone of height \(h\) and base radius \(r\) is at a height of \(\frac { 1 } { 4 } h\) above its base. The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius \(3 x\) and height \(6 y\), and removing a smaller cone, of base radius \(x\), with the same vertex. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575} Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is \(\frac { 18 } { 13 } y\).

\includegraphics{figure_7} A uniform solid cone has height \(1.2 \text{ m}\) and base radius \(0.5 \text{ m}\). A uniform object is made by drilling a cylindrical hole of radius \(0.2 \text{ m}\) through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is \(0.72 \text{ m}\) and that the volume of the cone removed by the drilling is \(0.0352\pi \text{ m}^3\). [4]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]

1. Find the distance of the centre of mass of the object from its base. [6]

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]

3 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]

3. \begin{figure}[h] \end{figure} A uniform solid cylinder has radius \(2 a\) and height \(h ( h > a )\).
A solid hemisphere of radius \(a\) is removed from the cylinder to form the vessel \(V\).
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The centre \(O\) of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.

1. Show that the centre of mass of \(V\) is \(\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }\) from \(O\). The vessel \(V\) is placed on a rough plane which is inclined at an angle \(\phi\) to the horizontal. The lower plane circular face of \(V\) is in contact with the inclined plane. Given that \(h = 5 a\), the plane is sufficiently rough to prevent \(V\) from slipping and \(V\) is on the point of toppling,
2. find, to three significant figures, the size of the angle \(\phi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).

1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
2. Find the value of \(y\) for which the container is about to topple.

7 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).

1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
2. Find the value of \(y\) for which the container is about to topple.

5 Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h] \end{figure}

1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_646_844_1452_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple. The ornament is now placed with its shaded face in contact with a rough horizontal surface. A force of magnitude \(P\) N, acting parallel to the planes of the L -shaped faces, is applied to one of the edges of the ornament, as shown in Fig. 5.3. The force is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the ornament and the surface is \(\mu\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-7_524_680_452_246} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} The value of \(P\) is gradually increased until the ornament is on the point of toppling but does not slide.
3. Determine the minimum value of \(\mu\).
4. Explain how your answer to part (c) would change if the angle between \(P\) and the horizontal was less than \(30 ^ { \circ }\).

\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).

1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]

The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.

1. Find the angle between \(CD\) and the vertical. [2]

A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

7 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-4_597_1011_251_566} \(A B C D E\) is the cross-section through the centre of mass of a uniform prism resting in equilibrium with \(D E\) on a horizontal surface. The cross-section has the shape of a square \(O B C D\) with sides of length \(a \mathrm {~m}\), from which a quadrant \(O A E\) of a circle of radius 1 m has been removed (see diagram).

1. Find the distance of the centre of mass of the prism from \(O\), giving the answer in terms of \(a , \pi\) and \(\sqrt { } 2\).
2. Hence show that $$3 a ^ { 2 } ( 2 - a ) < \frac { 3 } { 2 } \pi - 2$$ and verify that this inequality is satisfied by \(a = 1.68\) but not by \(a = 1.67\).

A thin uniform wire of mass \(12m\) is bent to form a right-angled triangle \(ABC\). The lengths of the sides \(AB\), \(BC\) and \(AC\) are \(3a\), \(4a\) and \(5a\) respectively. A particle of mass \(2m\) is attached to the triangle at \(B\) and a particle of mass \(3m\) is attached to the triangle at \(C\). The bent wire and the two particles form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium. Find the size of the angle between \(AB\) and the downward vertical. [10]

3 A triangular frame \(A B C\) consists of two uniform rigid rods each of length 0.8 m and weight 3 N , and a longer uniform rod of weight 4 N . The triangular frame has \(A B = B C\), and angle \(B A C =\) angle \(B C A = 30 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the frame from \(A C\). \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-2_722_335_1302_904} The vertex \(A\) of the frame is attached to a smooth hinge at a fixed point. The frame is held in equilibrium with \(A C\) vertical by a vertical force of magnitude \(F \mathrm {~N}\) applied to the frame at \(B\) (see diagram).
2. Calculate \(F\), and state the magnitude and direction of the force acting on the frame at the hinge.

2 Fig. 2.1 shows the positions of the points \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S } , \mathrm { T } , \mathrm { U } , \mathrm { V }\) and W which are at the vertices of a cube of side \(a\); Fig. 2.1 also shows coordinate axes, where O is the mid-point of PQ . \begin{figure}[h] \end{figure} An open box, A, is made from thin uniform material in the form of the faces of the cube with just the face TUVW missing.

1. Find the \(z\)-coordinate of the centre of mass of A . Strips made of a thin heavy material are now fixed to the edges TW, WV and VU of box A, as shown in Fig. 2.2. Each of these three strips has the same mass as one face of the box. This new object is B. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_488_476_1388_797} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Find the \(x\)-and \(z\)-coordinates of the centre of mass of B and show that the \(y\)-coordinate is \(\frac { 9 a } { 16 }\). Object B is now placed on a plane which is inclined at \(\theta\) to the horizontal. B is positioned so that face PQRS is on the plane with SR at right angles to a line of greatest slope of the plane and with PQ higher than SR , as shown in Fig. 2.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_237_284_2087_1555} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
3. Assuming that B does not slip, find \(\theta\) if B is on the point of tipping. B is now placed on a different plane which is inclined at \(30 ^ { \circ }\) to the horizontal. When B is released it accelerates down the plane at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the coefficient of friction between B and the inclined plane.

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B D E\), shown in Figure 1, has side \(A B\) of length \(2 a\) and side \(B D\) of length \(6 a\). The point \(C\) divides \(B D\) in the ratio 1:2 and the point \(F\) divides \(E A\) in the ratio \(1 : 2\). The rectangular lamina is folded along \(F C\) to produce the folded lamina \(L\), shown in Figure 2.

1. Show that the centre of mass of \(L\) is \(\frac { 16 } { 9 } a\) from \(E F\). The folded lamina, \(L\), is freely suspended from \(C\) and hangs in equilibrium.
2. Find the size of the angle between \(C F\) and the downward vertical.

4. \begin{figure}[h] \end{figure} The uniform square lamina \(A B C D\) shown in Figure 2 has sides of length 4a. The points \(E\) and \(F\), on \(D A\) and \(D C\) respectively, are both at a distance \(3 a\) from \(D\). The portion \(D E F\) of the lamina is folded through \(180 ^ { \circ }\) about \(E F\) to form the folded lamina \(A B C F E\) shown in Figure 3 below. \begin{figure}[h] \end{figure}

1. Show that the distance from \(A B\) of the centre of mass of the folded lamina is \(\frac { 55 } { 32 } a\).
   (6) The folded lamina is freely suspended from \(E\) and hangs in equilibrium.
2. Find the size of the angle between \(E D\) and the downward vertical.

2 The shaded region shown in Fig. 2.1 is cut from a sheet of thin rigid uniform metal; LBCK and EFHI are rectangles; EF is perpendicular to CK . The dimensions shown in the figure are in centimetres. The Oy and Oz axes are also shown. \begin{figure}[h] \end{figure}

1. Calculate the coordinates of the centre of mass of the metal shape referred to the axes shown in Fig. 2.1. The metal shape is freely suspended from the point H and hangs in equilibrium.
2. Calculate the angle that HI makes with the vertical. The metal shape is now folded along OJ , AD and EI to give the object shown in Fig. 2.2; LOJK, ABCD and IEFH are all perpendicular to OADJ; LOJK and ABCD are on one side of OADJ and IEFH is on the other side of it. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-3_542_929_1713_575} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Referred to the axes shown in Fig. 2.2, show that the \(x\)-coordinate of the centre of mass of the object is - 0.1 and find the other two coordinates of the centre of mass. The object is placed on a rough inclined plane with LOAB in contact with the plane. OL is parallel to a line of greatest slope of the plane with L higher than O . The object does not slip but is on the point of tipping about the edge OA .
4. Calculate the angle of OL to the horizontal.

\includegraphics{figure_2} A uniform lamina is in the form of a triangle \(ABC\) in which angle \(B\) is a right angle, \(AB = 9a\) and \(BC = 6a\). The point \(D\) is on \(BC\) such that \(BD = x\) (see diagram). The region \(ABD\) is removed from the lamina. The resulting shape \(ADC\) is placed with the edge \(DC\) on a horizontal surface and the plane \(ADC\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium. [6]

\includegraphics{figure_1} A uniform lamina \(ABCD\) is formed by removing the isosceles triangle \(ADC\) of height \(h\) metres, where \(h < 2\sqrt{3}\), from a uniform lamina \(ABC\) in the shape of an equilateral triangle of side 4 m, as shown in Figure 1. The centre of mass of \(ABCD\) is at \(D\).

1. Show that \(h = \sqrt{3}\) [7]

The weight of the lamina \(ABCD\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(AB\) vertical. The horizontal force acts in the vertical plane containing the lamina.

1. Find \(F\) in terms of \(W\). [4]

\includegraphics{figure_2} A uniform lamina is in the form of a triangle \(ABC\) in which angle \(B\) is a right angle, \(AB = 9a\) and \(BC = 6a\). The point \(D\) is on \(BC\) such that \(BD = x\) (see diagram). The region \(ABD\) is removed from the lamina. The resulting shape \(ADC\) is placed with the edge \(DC\) on a horizontal surface and the plane \(ADC\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium. [6]

1. Show that the distance \(d\) of the centre of mass of the toy from its lowest point \(O\) is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
2. Find \(h\) in terms of \(r\).
   (3)

2 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772} A uniform solid hemisphere, with centre \(O\) and radius 4 cm , is held so that a point \(P\) of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of \(70 ^ { \circ }\) with the horizontal. \(Q\) is the point on the axis of symmetry of the hemisphere which is vertically above \(P\). The diagram shows the vertical cross-section of the hemisphere which contains \(O , P\) and \(Q\).

1. Determine whether or not the centre of mass of the hemisphere is between \(O\) and \(Q\). The hemisphere is now released.
2. State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.

A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac{a}{2}\) from the centre and parallel to a diameter.

1. Show that the centre of gravity of the smaller cap from its plane face is \(\frac{7a}{40}\). [9 marks]

This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.

1. If the maximum value of \(\theta\) for which this is possible without the cap turning over is 30°, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical. [6 marks]

\includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]

\includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]

1. \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

\begin{figure}[h] \end{figure} Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where

• \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
• arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
• arc \(O C\) is one quarter of a circle of radius \(r\)
• all five pieces of wire lie in the same plane
  1. Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).

Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),

show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\) The framework is freely pivoted at \(A\).
The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\). Given that the weight of the framework is \(W\)

find \(F\) in terms of \(W\)

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 \(\frac { h } { 4 } ( k - 2 )\)

3. \begin{figure}[h] \end{figure} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle ABC , where \(\mathrm { AB } = \mathrm { AC } = 10 \mathrm {~cm}\) and \(\mathrm { BC } = 12 \mathrm {~cm}\), as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(B C\). The frame has total mass M . A particle of mass M is attached to the frame at the mid-point of BC . The frame is then freely suspended from B and hangs in equilibrium.
2. Find the size of the angle between BC and the vertical.