6.04c Composite bodies: centre of mass

6. Three equal uniform rods, each of mass \(m\) and length \(2 a\), form the sides of a rigid equilateral triangular frame \(A B C\). The frame is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the frame.

1. Show that the moment of inertia of the frame about \(L\) is \(6 m a ^ { 2 }\). The frame is held with \(A B\) horizontal and \(C\) below \(A B\), and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,
2. find the magnitude of the force exerted by the axis on the frame at \(A\) at the instant when the frame is released.

7. A pendulum consists of a uniform circular disc, of radius \(a\) and mass \(4 m\), whose centre is fixed to the end \(B\) of a uniform \(\operatorname { rod } A B\). The rod has mass \(3 m\) and length \(4 l\), where \(2 l > a\). The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about \(L\) is \(2 m \left( a ^ { 2 } + 40 l ^ { 2 } \right)\).

1. Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is held with \(B\) vertically above \(A\) and is then slightly displaced from rest. In the subsequent motion the midpoint of \(A B\) strikes a small peg, which is fixed at the same horizontal level as \(A\), and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is \(\omega\).
2. Show that \(\omega ^ { 2 } = \frac { 22 g l } { \left( a ^ { 2 } + 40 l ^ { 2 } \right) }\) Immediately after it strikes the peg, the angular speed of the pendulum is \(\frac { 1 } { 2 } \omega\).
3. Find, in terms of \(m , g , a\) and \(l\), the magnitude of the impulse exerted on the peg by the pendulum.
4. Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is \(\arcsin \frac { 1 } { 4 }\).
   \includegraphics[max width=\textwidth, alt={}]{1242d28a-a4bd-4754-ac49-9b48de95b880-24_2632_1830_121_121}

6.

1. Show by integration that the moment of inertia of a uniform disc, of mass \(m\) and radius \(a\), about an axis through the centre of disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
   (3 marks) \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4e874199-105a-460f-af7c-da0ef1603933-4_887_591_997_812}
   \end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). A uniform disc, of mass \(4 m\) and radius \(\frac { 1 } { 2 } a\), is attached to the rod with the centre of the disc lying on the rod a distance \(\frac { 3 } { 2 } a\) from \(A\). The rod lies in the plane of the disc, as shown in Fig. 1. The disc and rod together form a pendulum which is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the pendulum.
2. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 27 } { 2 } m a ^ { 2 }\). The pendulum makes small oscillations about its position of stable equilibrium.
3. Show that the motion of the pendulum is approximately simple harmonic, and find the period of the oscillations.
   (6 marks)
   

7. A uniform sphere, of mass \(m\) and radius \(a\), is free to rotate about a smooth fixed horizontal axis \(L\) which forms a tangent to the sphere. The sphere is hanging in equilibrium below the axis when it receives an impulse, causing it to rotate about \(L\) with an initial angular velocity of \(\sqrt { \frac { 18 g } { 7 a } }\). Show that, when the sphere has turned through an angle \(\theta\),

1. the angular speed \(\omega\) of the sphere is given by \(\omega ^ { 2 } = \frac { 2 g } { 7 a } ( 4 + 5 \cos \theta )\),
2. the angular acceleration of the sphere has magnitude \(\frac { 5 g } { 7 a } \sin \theta\).
3. Hence find the magnitude of the force exerted by the axis on the sphere when the sphere comes to instantaneous rest for the first time. END

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.

6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).

1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.

7 The vertices of a uniform triangular lamina, which is in the \(x - y\) plane, are at the origin and the points \(( 20,60 )\) and \(( 100,0 )\).

1. Determine the coordinates of the lamina's centre of mass. Fig. 7.1 shows a uniform lamina consisting of a triangular section and two identical rectangular sections. The coordinates of some of the vertices of the lamina are given in Fig. 7.1. The rectangular sections are then folded at right-angles to the triangular section, to give the rigid three-dimensional object illustrated in Fig. 7.2. Two of the edges, \(E _ { 1 }\) and \(E _ { 2 }\), are marked on both figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_933_739_799_164} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_924_725_808_1133} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
2. Show that the \(x\)-coordinate of the centre of mass of the folded object is 43.6, and determine the \(y\) - and \(z\)-coordinates.
3. Determine whether it is possible for the folded object to rest in equilibrium with edges \(E _ { 1 }\) and \(E _ { 2 }\) in contact with a horizontal surface.

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 Fig. 3.1 shows the curve with equation \(y = x ^ { 2 } + 3\). The region \(R\), shown shaded, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). A uniform solid of revolution S is formed by rotating the region R through \(2 \pi\) about the \(x\)-axis. The volume of \(S\) is \(\frac { 202 } { 5 } \pi\). \begin{figure}[h] \end{figure} \section*{(a) In this question you must show detailed reasoning.} Show that the \(x\)-coordinate of the centre of mass of S is \(\frac { 395 } { 303 }\). S is fixed to a cylinder of base radius 3 units and height 2 units to form the uniform solid D . The smaller circular face of S is joined to the top circular face of the cylinder, as shown in Fig. 3.2. \begin{figure}[h] \end{figure} (b) Find the distance of the centre of mass of D from its smaller circular face. D is placed with its smaller circular face in contact with a rough plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is given that D does not slip.
(c) Determine whether D topples.

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}

5 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 120,0 )\), \(( 60,90 )\) and \(( 30,90 )\) respectively (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-5_561_720_404_242} The centre of mass of the lamina lies at ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 54\) and determine the value of \(\bar { y }\). The lamina is placed horizontally so that it rests on three supports, whose points of contact are at \(\mathrm { B } , \mathrm { C }\) and D , where D lies on the edge OA and has coordinates \(( d , 0 )\).
2. Determine the range of values of \(d\) for the lamina to rest in equilibrium. It is now given that \(d = 63\), and that the lamina has a weight of 100 N .
3. Determine the forces exerted on the lamina by each of the supports at \(\mathrm { B } , \mathrm { C }\) and D .

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.

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 \(30 ^ { \circ }\).

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.

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 \(k\).

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 \(\frac { 18 } { 13 } y\).

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 \(A Q\).

2.

1. Prove that the centre of mass of a uniform solid cone of height \(h\) and base radius \(b\) is at a height of \(\frac { 1 } { 4 } h\) above its base.
2. A uniform solid cone \(C _ { 1 }\) has height 3 m and base radius 2 m . A smaller cone \(C _ { 2 }\) of height 2 m and base radius 1 m is contained symmetrically inside \(C _ { 1 }\). The bases of \(C _ { 1 }\) and \(C _ { 2 }\) have a common centre and the axis of \(C _ { 2 }\) is part of the axis of \(C _ { 1 }\). If \(C _ { 2 }\) is removed from \(C _ { 1 }\), show that the centre of mass of the remaining solid is at a distance of \(\frac { 11 } { 5 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\).
3. The remaining solid is suspended from a string which is attached to a point on the outer curved surface at a distance of \(\frac { 1 } { 3 } \sqrt { 13 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\). Given that the axis of symmetry is inclined at an angle of \(\alpha\) to the vertical, find \(\tan \alpha\).