6.04c Composite bodies: centre of mass

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

4 \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4. \begin{figure}[h] \end{figure} The uniform lamina \(O B C\) is one quarter of a circular disc with centre \(O\) and radius 4 m . The points \(A\) and \(D\), on \(O B\) and \(O C\) respectively, are 3 m from \(O\). The uniform lamina \(A B C D\), shown shaded in Figure 1, is formed by removing the triangle \(O A D\) from \(O B C\). Given that the centre of mass of one quarter of a uniform circular disc of radius \(r\) is at a distance \(\frac { 4 \sqrt { 2 } } { 3 \pi } r\) from the centre of the disc,

1. find the distance of the centre of mass of the lamina \(A B C D\) from \(A D\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
2. Find, to the nearest degree, the angle between \(D C\) and the downward vertical.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-09_915_1269_118_356} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}

6 \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-3_540_878_989_632} A uniform lamina \(A B C D E\) of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the diagram.

1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina. The lamina is freely suspended from a hinge at \(B\).
2. Calculate the angle that \(A B\) makes with the vertical. The lamina is now held in a position such that \(B D\) is horizontal. This is achieved by means of a string attached to \(D\) and to a fixed point 15 cm directly above the hinge at \(B\).
3. Calculate the tension in the string.

8

1. A uniform semicircular lamina has radius 4 cm . Show that the distance from its centre to its centre of mass is 1.70 cm , correct to 3 significant figures.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_429_947_405_640} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A model bridge is made from a uniform rectangular board, \(A B C D\), with a semicircular section, \(E F G\), removed. \(O\) is the mid-point of \(E G\). \(A B = 8 \mathrm {~cm} , B C = 20 \mathrm {~cm} , A O = 12 \mathrm {~cm}\) and the radius of the semicircle is 4 cm (see Fig. 1).
   1. Show that the distance from \(A B\) to the centre of mass of the model is 9.63 cm , correct to 3 significant figures.
   2. Calculate the distance from \(A D\) to the centre of mass of the model.
   3. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_572_945_1416_641} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} The model bridge is smoothly pivoted at \(A\) and is supported in equilibrium by a vertical wire attached to \(D\). The weight of the model is 15 N and \(A D\) makes an angle of \(10 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the tension in the wire.

3 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_828_476_1338_836} A door is modelled as a lamina \(A B C D E\) consisting of a uniform rectangular section \(A B D E\) of weight 60 N and a uniform semicircular section \(B C D\) of weight 10 N and radius \(40 \mathrm {~cm} . A B\) is 200 cm and \(A E\) is 80 cm . The door is freely hinged at \(F\) and \(G\), where \(G\) is 30 cm below \(B\) and \(F\) is 30 cm above \(A\) (see diagram).

1. Find the magnitudes and directions of the horizontal components of the forces on the door at each of \(F\) and \(G\).
2. Calculate the distance from \(A E\) to the centre of mass of the door.

3 \begin{figure}[h] \end{figure} A uniform conical shell has mass 0.2 kg , height 0.3 m and base diameter 0.8 m . A uniform hollow cylinder has mass 0.3 kg , length 0.7 m and diameter 0.8 m . The conical shell is attached to the cylinder, with the circumference of its base coinciding with one end of the cylinder (see Fig. 1).

1. Show that the distance of the centre of mass of the combined object from the vertex of the conical shell is 0.47 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_497_572_1836_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The combined object is freely suspended from its vertex and is held with its axis horizontal. This is achieved by means of a wire attached to a point on the circumference of the base of the conical shell. The wire makes an angle of \(80 ^ { \circ }\) with the slant edge of the conical shell (see Fig. 2).
2. Calculate the tension in the wire.

1 \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_378_471_260_836} A uniform square frame \(A B C D\) has sides of length 0.6 m . The side \(A D\) is removed from the frame, and the open frame \(A B C D\) is attached at \(A\) to a fixed point (see diagram).

1. Calculate the distance of the centre of mass of the open frame from \(A\). The open frame rotates about \(A\) in the plane \(A B C D\) with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the speed of the centre of mass of the open frame.

5 A uniform solid is made of a hemisphere with centre \(O\) and radius 0.6 m , and a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere and a plane face of the cylinder coincide. (The formula for the volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).)

1. Show that the distance of the centre of mass of the solid from \(O\) is 0.09 m .
2. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-03_636_1036_982_593} The solid is placed with the curved surface of the hemisphere on a rough horizontal surface and the axis inclined at \(45 ^ { \circ }\) to the horizontal. The equilibrium of the solid is maintained by a horizontal force of 2 N applied to the highest point on the circumference of its plane face (see diagram). Calculate
   1. the mass of the solid,
   2. the set of possible values of the coefficient of friction between the surface and the solid.

2 \begin{figure}[h] \end{figure} A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
[0pt] [The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

1. Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\). The toy is placed on a horizontal surface with the hemisphere in contact with the surface. The toy is released from rest from the position in which the common plane circular face is vertical (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_193_670_1827_699} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. Find the set of values of \(\alpha\) such that the toy moves to the upright position.

7 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .

1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.

3 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).

1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
2. Calculate the tensions in the strings.

5 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-3_657_549_1219_799} A uniform lamina \(A B C D E\) consists of a square and an isosceles triangle. The square has sides of 18 cm and \(B C = C D = 15 \mathrm {~cm}\) (see diagram).

1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
2. The lamina is freely suspended from \(B\). Calculate the angle that \(B D\) makes with the vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_441_1355_265_394} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A light inextensible string of length 1 m passes through a small smooth hole \(A\) in a fixed smooth horizontal plane. One end of the string is attached to a particle \(P\), of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle \(Q\), of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).

5 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(\mathrm { y } = 6 + \sin \mathrm { x }\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 5.1). The centre of mass of \(L\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \begin{figure}[h] \end{figure}

1. Show that \(\bar { X } = 2.36\), correct to 3 significant figures.
2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The cover of the book weighs 6 N . \(A\) is the point on the cover with coordinates \(( 3 , \bar { y } )\) and \(B\) is the point on the cover with coordinates \(( 5 , \bar { y } )\). A small badge of weight 2 N is attached to the cover at \(A\). The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P N\) acting at \(B\) perpendicular to the cover (see Fig. 5.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.

7 Fig. 7.1 shows a uniform lamina in the shape of a sector of a circle of radius \(r\) and angle \(2 \theta\) where \(\theta\) is in radians. The sector consists of a triangle \(O A B\) and a segment bounded by the chord \(A B\). \begin{figure}[h] \end{figure}

1. Explain why the centre of mass of the segment lies on the radius through the midpoint of \(A B\).
2. Show that the distance of the centre of mass of the segment from \(O\) is \(\frac { 2 r \sin ^ { 3 } \theta } { 3 ( \theta - \sin \theta \cos \theta ) }\). A uniform circular lamina of radius 5 units is placed with its centre at the origin, \(O\), of an \(x - y\) coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is \(O\). The centre of the straight edge of the segment has coordinates ( 0,2 ) and this edge is perpendicular to the \(y\)-axis (see Fig. 7.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-6_766_757_1400_244} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Find the \(y\)-coordinate of the centre of mass of the component, giving your answer correct to 3 significant figures. \(C\) is the point on the component with coordinates ( 0,5 ). The component is now placed horizontally and supported only at \(O\). A particle of mass \(m \mathrm {~kg}\) is placed on the component at \(C\) and the component and particle are in equilibrium.
4. Find the mass of the component in terms of \(m\).

8 A rectangular lamina of mass \(M\) has vertices at the origin \(O ( 0,0 ) , A ( 24 a , 0 ) , B ( 24 a , 6 a )\) and \(C ( 0,6 a )\), where \(a\) is a positive constant. A small object \(P\) of mass \(m\) is attached to the lamina at the point ( \(x , y\) ). The centre of mass of the system consisting of the lamina and \(P\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \(P\) is modelled as a particle and the lamina is modelled as being uniform.

1. Show that \(x = \frac { 12 M a + m x } { M + m }\).
2. Find a corresponding expression for \(\bar { y }\). The lamina, with \(P\) no longer attached, is placed on a horizontal rectangular table, with its sides parallel to the edges of the table, and partly overhanging the edges of the table, as shown in the diagram. The corner of the table is at the point ( \(6 a , 2 a\) ). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-6_538_1431_849_246} When \(P\) is placed on the lamina at \(O\), the lamina topples over one of the edges of the table.
3. Show that \(\mathrm { m } > \frac { 1 } { 2 } \mathrm { M }\). The lamina is now put back on the table in the same position as before. \(P\) is placed at the point \(( 12 a , 6 a )\) on the smooth upper surface of the lamina, and is projected towards \(O\). At a subsequent instant during the motion, \(P\) is at the point (12ak, 6ak) where \(0 < k < 1\).
4. Assuming that the lamina has not yet toppled, find, in terms of \(M\) and \(m\), the value of \(k\) for which the centre of mass of the system lies on the table edge parallel to \(O C\).
5. For the case \(\mathrm { m } = \frac { 3 } { 2 } \mathrm { M }\), determine which table edge the lamina topples over.

6. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\) and rolled along the plank to \(B\) without either string breaking.

1. Explain, with the aid of a sketch-graph, how the tension in the string at \(A\) varies with \(x\), the distance of the rock from \(A\).
2. Show that \(M \leq 15\). The first rock is removed and a second rock of mass 20 kg is placed on the plank.
3. Find the fraction of the plank on which the rock can be placed without one of the strings breaking.

4 The diagram shows a uniform lamina \(A B C D E F G H\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-3_346_933_1123_577}

1. Explain why the centre of mass is 25 cm from \(A H\).
2. Show that the centre of mass is 4.375 cm from \(H G\).
3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
4. Explain, briefly, how you have used the fact that the lamina is uniform.

4 A uniform rectangular lamina \(A B C D\) has a mass of 5 kg . The side \(A B\) has length 60 cm and the side \(B C\) has length 30 cm . The points \(P , Q , R\) and \(S\) are the mid-points of the sides, as shown in the diagram below. A uniform triangular lamina \(S R D\), of mass 4 kg , is fixed to the rectangular lamina to form a shop sign. The centre of mass of the triangular lamina \(S R D\) is 10 cm from the side \(A D\) and 5 cm from the side \(D C\). \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-08_613_1086_660_518}

1. Find the distance of the centre of mass of the shop sign from \(A D\).
2. Find the distance of the centre of mass of the shop sign from \(A B\).
3. The shop sign is freely suspended from \(P\). Find the angle between \(A B\) and the horizontal when the shop sign is in equilibrium.
4. To ensure that the side \(A B\) is horizontal when the shop sign is freely suspended from point \(P\), a particle of mass \(m \mathrm {~kg}\) is attached to the shop sign at point \(B\). Calculate \(m\).
5. Explain how you have used the fact that the rectangular lamina \(A B C D\) is uniform in your solution to this question.
   (1 mark)
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-10_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-11_2486_1714_221_153}