6.04b Find centre of mass: using symmetry

5 Four identical uniform rods, each of mass \(m\) and length \(2 a\), are rigidly joined to form a square frame \(A B C D\). Show that the moment of inertia of the frame about an axis through \(A\) perpendicular to the plane of the frame is \(\frac { 40 } { 3 } m a ^ { 2 }\). The frame is suspended from \(A\) and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through \(A\). When the frame is at rest with \(C\) vertically below \(A\), it is given an angular velocity \(\sqrt { } \left( \frac { 6 g } { 5 a } \right)\). Find the angular velocity of the frame when \(A C\) makes an angle \(\theta\) with the downward vertical through \(A\). When \(A C\) is horizontal, the speed of \(C\) is \(k \sqrt { } ( g a )\). Find the value of \(k\) correct to 3 significant figures.

1 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_553_435_258_854} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).

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.

2 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_465_643_495_749} A uniform right-angled triangular lamina \(A B C\) with sides \(A B = 12 \mathrm {~cm} , B C = 9 \mathrm {~cm}\) and \(A C = 15 \mathrm {~cm}\) is freely suspended from a hinge at its vertex \(A\). The lamina has mass 2 kg and is held in equilibrium with \(A B\) horizontal by means of a string attached to \(B\). The string is at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram). Calculate the tension in the string.

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.

8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .

1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
2. Determine the natural length of the elastic string.

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.

4. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of mass 50 kg and length 5 m which overhangs a river by 2 m . When a boy of mass 20 kg stands at \(A\), his sister can walk to within 0.3 m of \(B\), at which point the plank is in limiting equilibrium.

1. What is the mass of the girl?
2. Find the smallest extra weight which must be placed at \(A\) to enable the girl to walk right to the end \(B\).
3. How have you used the fact that the plank is uniform?

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}

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.

2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}

1. Explain why the centre of mass of the lamina lies on \(P Q\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.

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}

2 The diagram shows four particles, \(A , B , C\) and \(D\), which are fixed in a horizontal plane which contains the \(x\) - and \(y\)-axes, as shown. Particle \(A\) has mass 2 kg and is attached at the point ( 9,6 ).
Particle \(B\) has mass 3 kg and is attached at the point ( 2,4 ).
Particle \(C\) has mass 8 kg and is attached at the point \(( 3,8 )\).
Particle \(D\) has mass 7 kg and is attached at the point \(( 6,11 )\). \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-2_748_774_1402_625} Find the coordinates of the centre of mass of the four particles.