6.04b Find centre of mass: using symmetry

2 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\), and the curve \(y = k x ^ { 2 }\), where \(k\) is a positive constant, is occupied by a uniform lamina.

1. Find the exact \(x\)-coordinate of the centre of mass of the lamina.
2. Given that the \(x\) - and \(y\)-coordinates of the centre of mass of the lamina are equal, find the exact value of \(k\).

4 A uniform lamina of mass \(m\) is in the shape of a sector of a circle of radius \(a\) and angle \(\frac { 1 } { 3 } \pi\). It can rotate freely in a vertical plane about a horizontal axis perpendicular to the lamina through its vertex O .

1. Show by integration that the moment of inertia of the lamina about the axis is \(\frac { 1 } { 2 } m a ^ { 2 }\).
2. State the distance of the centre of mass of the lamina from the axis. The lamina is released from rest when one of the straight edges is horizontal as shown in Fig. 4.1. After time \(t\), the line of symmetry of the lamina makes an angle \(\theta\) with the downward vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_257_441_1475_322} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_380_732_1635_1014} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
3. Show that \(\dot { \theta } ^ { 2 } = \frac { 4 g } { \pi a } ( 2 \cos \theta + 1 )\).
4. Find the greatest speed attained by any point on the lamina.
5. Find an expression for \(\ddot { \theta }\) in terms of \(\theta , a\) and \(g\). The lamina strikes a fixed peg at A where \(\mathrm { AO } = \frac { 3 } { 4 } a\) and is horizontal, as shown in Fig. 4.2. The collision reverses the direction of motion of the lamina and halves its angular speed.
6. Find the magnitude of the impulse that the peg gives to the lamina.
7. Determine the maximum value of \(\theta\) in the subsequent motion.

4 A solid cylinder of radius \(a \mathrm {~m}\) and length \(3 a \mathrm {~m}\) has density \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) given by \(\rho = k \left( 2 + \frac { x } { a } \right)\) where \(x \mathrm {~m}\) is the distance from one end and \(k\) is a positive constant. The mass of the cylinder is \(M \mathrm {~kg}\) where \(M = \frac { 21 } { 2 } \pi a ^ { 3 } k\). Let A and B denote the circular faces of the cylinder where \(x = 0\) and \(x = 3 a\), respectively.

1. Show by integration that the moment of inertia, \(I _ { \mathrm { A } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face A is given by \(I _ { \mathrm { A } } = \frac { 109 } { 28 } M a ^ { 2 }\).
2. Show that the centre of mass of the cylinder is \(\frac { 12 } { 7 } a \mathrm {~m}\) from A .
3. Using the parallel axes theorem, or otherwise, show that the moment of inertia, \(I _ { \mathrm { B } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face B is given by \(I _ { \mathrm { B } } = \frac { 73 } { 28 } M a ^ { 2 }\). You are now given that \(M = 4\) and \(a = 0.7\). The cylinder is at rest and can rotate freely about a horizontal axis which is a diameter of the face B as shown in Fig. 4. It is struck at the bottom of the curved surface by a small object of mass 0.2 kg which is travelling horizontally at speed \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical plane which is both perpendicular to the axis of rotation and contains the axis of symmetry of the cylinder. The object sticks to the cylinder at the point of impact. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-4_606_435_1087_817} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Find the initial angular speed of the combined object after the collision. \section*{END OF QUESTION PAPER}

4. Show, using integration, that the moment of inertia of a uniform solid right circular cone of mass \(M\), height \(h\) and base radius \(a\), about an axis through the vertex, parallel to the base, is $$\frac { 3 M } { 20 } \left( a ^ { 2 } + 4 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform circular disc, of radius \(r\) and mass \(m\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]

5. \begin{figure}[h] \end{figure} A uniform circular lamina has radius \(2 a\) and centre \(C\). The points \(P , Q , R\) and \(S\) on the lamina are the vertices of a square with centre \(C\) and \(C P = a\). Four circular discs, each of radius \(\frac { a } { 2 }\), with centres \(P , Q , R\) and \(S\), are removed from the lamina. The remaining lamina forms a template \(T\), as shown in Figure 1. The radius of gyration of \(T\) about an axis through \(C\), perpendicular to \(T\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }\) The template \(T\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(T\) and passes through a point on its outer rim.
2. Write down an equation of rotational motion for \(T\) and deduce that the period of small oscillations of \(T\) about its stable equilibrium position is $$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$

2. A uniform square lamina \(S\) has side \(2 a\). The radius of gyration of \(S\) about an axis through a vertex, perpendicular to \(S\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 8 a ^ { 2 } } { 3 }\). The lamina \(S\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(S\) and passes through a vertex.
2. By writing down an equation of rotational motion for \(S\), find the period of small oscillations of \(S\) about its position of stable equilibrium.

5. \begin{figure}[h] \end{figure} A uniform triangular lamina \(A B C\), of mass \(M\), has \(A B = A C\) and \(B C = 2 a\). The mid-point of \(B C\) is \(D\) and \(A D = h\), as shown in Figure 1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), perpendicular to the plane of the lamina, is $$\frac { M } { 6 } \left( a ^ { 2 } + 3 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform rod, of length \(2 l\) and mass \(m\), about an axis through its midpoint and perpendicular to the rod, is \(\frac { 1 } { 3 } m l ^ { 2 }\).]

4. A uniform solid sphere has mass \(M\) and radius \(a\). Prove, using integration, that the moment of inertia of the sphere about a diameter is \(\frac { 2 M a ^ { 2 } } { 5 }\) [0pt] [You may assume without proof that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\frac { 1 } { 2 } m r ^ { 2 }\).]

8. \begin{figure}[h] \end{figure} A uniform circular disc of radius \(2 a\) has centre \(O\). The points \(P , Q , R\) and \(S\) on the disc are the vertices of a square with centre \(O\) and \(O P = a\). Four circular holes, each of radius \(\frac { a } { 2 }\), and with centres \(P , Q , R\) and \(S\), are drilled in the disc to produce the lamina \(L\), shown shaded in Figure 1. The mass of \(L\) is \(M\).

1. Show that the moment of inertia of \(L\) about an axis through \(O\), and perpendicular to the plane of \(L\), is \(\frac { 55 M a ^ { 2 } } { 24 }\) The lamina \(L\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(L\) and which passes through a point \(A\) on the circumference of \(L\). At time \(t , A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta\)
3. Hence find the period of small oscillations of \(L\) about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to \(A O\), of the force exerted on \(L\) by the axis is \(X\).
4. Find \(X\) in terms of \(M , g\) and \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}

4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle \(\alpha\) with \(\mathrm { O } y\), where \(\sin \alpha = \frac { 7 } { 25 }\). The rod and lamina are rigidly attached at E, the mid-point of CD.
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg . \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Find,
   1. the \(y\)-coordinate of the centre of mass of the shovel,
   2. the \(z\)-coordinate of the centre of mass of the shovel. The shovel is freely suspended from O and hangs in equilibrium.
2. Calculate the angle that OE makes with the vertical.

3 Fig. 3.1 shows a thin rectangular frame ABCD , with part of it filled by a triangular lamina ABD . \(\mathrm { AD } = 30 \mathrm {~cm}\) and \(\mathrm { AB } = x \mathrm {~cm}\). Together they form the composite structure S . The centre of mass of \(S\) lies at a point \(M , 16.5 \mathrm {~cm}\) from \(A D\) and 11.7 cm from \(A B\). \begin{figure}[h] \end{figure} The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .

1. Show that the triangular lamina has a mass of 3.3 kg .
2. Determine the value of \(x\), correct to \(\mathbf { 3 }\) significant figures. One end of a light inextensible string is attached to S at D . The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude \(Q N\) is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S . The string is taut and lies in the same plane as S at an angle \(\phi\) to the downward vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. By taking moments about D , show that \(Q = 50.5\), correct to 3 significant figures.
4. Determine, in degrees, the value of \(\phi\).

5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h] \end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
   \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
   \end{figure} The weight of S is \(W \mathrm {~N}\).
3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .

6 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(A , B\) and \(C\) have coordinates \(( 12,0 ) , ( 12 + p , q )\) and \(( 0 , q )\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-7_536_917_349_239}

1. Determine, in terms of \(p\) and \(q\), the coordinates of the centre of mass of OABC . The point D has coordinates \(( 7.6 , q )\). When OABC is suspended from D , the lamina hangs in equilibrium with BC horizontal.
2. Determine the value of \(p\). When OABC is suspended from C, the lamina hangs in equilibrium with BC at an angle of \(35 ^ { \circ }\) to the downward vertical.
3. Determine the value of \(q\), giving your answer correct to \(\mathbf { 3 }\) significant figures.

2 A triangular lamina, ABC , is cut from a piece of thin uniform plane sheet metal. The dimensions of ABC are shown in Fig. 2. \begin{figure}[h] \end{figure} This piece of metal is freely suspended from a string attached to C and hangs in equilibrium. Calculate the angle of BC with the downward vertical, giving your answer in degrees.

5 In this question, all coordinates refer to the axes shown in Fig. 5.1. Fig. 5.1 shows a system of four particles with masses \(4 m , 3 m , m\) and \(2 m\) at the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . These points have coordinates \(( - 3,4 ) , ( 0,0 ) , ( 2,0 )\) and \(( 5,4 )\). \begin{figure}[h] \end{figure}

1. Calculate the coordinates of the centre of mass of the system of particles. A thin uniform rigid wire of mass \(12 m\) connects the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D with straight line sections, as shown in Fig. 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-5_460_903_1338_573} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Calculate the coordinates of the centre of mass of the wire. The particles at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are now fixed to the wire to form a rigid object, \(R\).
3. Calculate the \(x\)-coordinate of the centre of mass of \(R\).

5 Fig. 5 shows the curve with equation \(y = - x ^ { 2 } + 4 x + 2\).
The curve intersects the \(x\)-axis at P and Q . The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 4\) is occupied by a uniform lamina L . The horizontal base of L is OA , where A is the point \(( 4,0 )\). \begin{figure}[h] \end{figure}

1. 1. Explain why the centre of mass of L lies on the line \(x = 2\).
   2. In this question you must show detailed reasoning. Find the \(y\)-coordinate of the centre of mass of \(L\).
2. L is freely suspended from A . Find the angle AO makes with the vertical. The region bounded by the curve and the \(x\)-axis is now occupied by a uniform lamina M . The horizontal base of M is PQ.
3. Explain how the position of the centre of mass of M differs from the position of the centre of mass of \(L\).

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.

6 Fig. 6.1 shows a light rod ABC , of length 60 cm , where B is the midpoint of AC . Particles of masses \(3.5 \mathrm {~kg} , 1.4 \mathrm {~kg}\) and 2.1 kg are attached to \(\mathrm { A } , \mathrm { B }\) and C respectively. \begin{figure}[h] \end{figure} The centre of mass is located at a point G along the rod.

1. Determine the distance AG . Two light inextensible strings, each of length 40 cm , are attached to the rod, one at A , the other at C. The other ends of these strings are attached to a fixed point D. The rod is allowed to hang in equilibrium.
2. Determine the angle AD makes with the vertical. The two strings are now replaced by a single light inextensible string of length 80 cm . One end of the string is attached to A and the other end of the string is attached to C. The string passes over a smooth peg fixed at D. The rod hangs in equilibrium, but is not vertical, as shown in Fig. 6.2. Fig. 6.2
3. Explain why angle ADG and angle CDG must be equal.
4. Determine the tension in the string.

6 In this question you may use the fact that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).
Fig. 6.1 shows a container in the shape of an open-topped cylinder. The cylinder has height 18 cm and radius 4 cm . The curved surface and the base can be modelled as uniform laminae with the same mass per unit area. The container rests on a horizontal surface. \begin{figure}[h] \end{figure}

1. Show that the centre of mass of the container lies 8.1 cm above its base. The mass of the container is 400 grams. Water is poured into the container to reach a height of \(h \mathrm {~cm}\) above the base. The centre of mass of the combined container and water lies \(y \mathrm {~cm}\) above the base. Water has a density of 1 gram per \(\mathrm { cm } ^ { 3 }\).
2. In this question you must show detailed reasoning. By formulating an expression for \(y\) in terms of \(h\), determine the value of \(h\) for which \(y\) is lowest. More water is now poured into the container. A sphere of radius 3 cm is placed into the container, where it sinks to the bottom. The surface of the water is now 4.5 cm from the top of the container, as shown in Fig. 6.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-6_432_355_2001_255}
   \end{figure}
3. Show that the centre of mass of the water in the container lies 7.5 cm above the base of the container. The sphere has a density of 4 grams per \(\mathrm { cm } ^ { 3 }\).
   The centre of mass of the combined container, water and sphere lies \(z \mathrm {~cm}\) above the base.
4. Determine the value of \(z\). \section*{END OF QUESTION PAPER}

1 A uniform solid rectangular prism has cross-section with width \(w \mathrm {~cm}\) and height 24 cm . Another uniform solid prism has cross-section in the shape of an isosceles triangle with width \(w \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The prisms are both placed with their axes vertical on a rough horizontal plane (see Fig. 1.1, which shows the cross-sections through the centres of mass of both solids). \begin{figure}[h] \end{figure} The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha\), when both solids topple simultaneously.

1. Determine the value of \(h\). It is given that \(w = 12\).
2. Determine the value of \(\alpha\). Both prisms are made from the same material and are of uniform density. The triangular prism is now placed on top of the rectangular prism to form a composite body C such that the base of the triangular prism coincides with the top of the rectangular prism. A cross-section of C is shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_777_439_1784_258} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Determine the height of the centre of mass of C from its base.

2 The diagram shows a system of three particles of masses \(3 m , 5 m\) and \(2 m\) situated in the \(x - y\) plane at the points \(\mathrm { A } ( 1,2 ) , \mathrm { B } ( 2 , - 2 )\) and \(\mathrm { C } ( 5,3 )\) respectively.
Determine the coordinates of the centre of mass of the system of particles.

2. A metal sign is formed by removing triangle \(B C D\) from a rectangular lamina \(A C E F\) made of uniform material, and adding a quarter circle XYZ, made of the same uniform material, with a particle attached to its vertex at \(Y\). The sign is supported by two light chains fixed at \(E\) and \(F\). The quarter circle has radius 24 cm and the particle at \(Y\) has a mass equal to half of that of the removed triangle. \(X D\) is parallel to \(A C\) and \(B Z\) is parallel to \(A F\). The dimensions, in cm , are as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-3_885_636_712_715}

1. Calculate the distance of the centre of mass of the sign from
   1. \(A F\),
   2. \(A C\).
2. The support at \(F\) comes loose so that the sign is freely suspended at \(E\) by one chain alone. Given that it then hangs in equilibrium, calculate the angle that \(E F\) makes with the vertical.

5. (a) Show, by integration, that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance of \(\frac { 3 r } { 8 }\) from the plane face.
(b) The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is \(3 r \mathrm {~cm}\), where \(r\) represents the common radius of the hemisphere and the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858} Given that the density of the hemisphere is \(50 \%\) more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.

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 \(k\).

4. The diagram shows three light rods \(A B , B C\) and \(C A\) rigidly joined together so that \(A B C\) is a right-angled triangle with \(A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}\) and \(\widehat { A B } = 90 ^ { \circ }\). The rods support a uniform lamina, of density \(2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), in the shape of a quarter circle \(A D E\) with radius 12 cm and centre at the vertex \(A\). Three particles are attached to \(B C\) : one at \(B\), one at \(C\) and one at \(F\), the midpoint of \(B C\). The masses at \(C , F\) and \(B\) are \(50 m \mathrm {~kg} , 30 m \mathrm {~kg}\) and \(20 m \mathrm {~kg}\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}

1. Calculate the distance of the centre of mass of the system from
   1. \(A C\),
   2. \(A B\).
2. When the system is freely suspended from a point \(P\) on \(A C\), it hangs in equilibrium with \(A B\) vertical. Write down the length \(A P\).
3. When the system is freely suspended from a point \(Q\) on \(A D\), it hangs in equilibrium with \(Q B\) making an angle of \(60 ^ { \circ }\) with the vertical. Find the distance \(A Q\).