6.04c Composite bodies: centre of mass

2. \begin{figure}[h] \end{figure} A uniform solid consists of a right circular cone of radius \(r\) and height \(k r\), where \(k > \sqrt { } 3\), fixed to a hemisphere of radius \(r\). The centre of the plane face of the hemisphere is \(O\) and this plane face coincides with the base of the cone, as shown in Figure 1.

1. Show that the distance of the centre of mass of the solid from \(O\) is $$\frac { \left( k ^ { 2 } - 3 \right) r } { 4 ( k + 2 ) }$$ The point \(A\) lies on the circumference of the base of the cone. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. The angle between \(A O\) and the vertical is \(\theta\), where \(\tan \theta = \frac { 11 } { 14 }\)
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} A toy is formed by joining a uniform solid right circular cone, of base radius \(3 r\) and height \(4 r\), to a uniform solid cylinder, also of radius \(3 r\) and height \(4 r\). The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is \(O\).

1. Find the distance of the centre of mass of the toy from \(O\). The point \(A\) lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from \(A\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical. The toy is placed with the curved surface of the cone on horizontal ground.
3. Determine whether the toy will topple.
   (4)

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

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

4. \begin{figure}[h] \end{figure} A uniform solid hemisphere, of radius \(6 a\) and centre \(O\), has a solid hemisphere of radius \(2 a\), and centre \(O\), removed to form a bowl \(B\) as shown in Figure 3.

1. Show that the centre of mass of \(B\) is \(\frac { 30 } { 13 } a\) from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The bowl \(B\) is fixed to a plane face of a uniform solid cylinder made from the same material as \(B\). The cylinder has radius \(2 a\) and height \(6 a\) and the combined solid \(S\) has an axis of symmetry which passes through \(O\), as shown in Figure 4.
2. Show that the centre of mass of \(S\) is \(\frac { 201 } { 61 } a\) from \(O\). The plane surface of the cylindrical base of \(S\) is placed on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent slipping.
3. Determine whether or not \(S\) will topple. \section*{
   \includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
   }

4. \begin{figure}[h] \end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2 l\) and the base of the cone has radius \(l\).

1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_460_588_1254_676} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
2. Find the value of \(\theta\).

3. \begin{figure}[h] \end{figure} A solid consists of a uniform solid right cylinder of height \(5 l\) and radius \(3 l\) joined to a uniform solid hemisphere of radius \(3 l\). The plane face of the hemisphere coincides with a circular end of the cylinder and has centre \(O\), as shown in Figure 2. The density of the hemisphere is twice the density of the cylinder.

1. Find the distance of the centre of mass of the solid from \(O\).
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_618_807_1327_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The solid is now placed with its circular face on a plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.
2. Find the value of \(\theta\).

6.

1. A uniform lamina is in the shape of a quadrant of a circle of radius \(a\). Show, by integration, that the centre of mass of the lamina is at a distance of \(\frac { 4 a } { 3 \pi }\) from each of its straight edges. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-10_809_802_484_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A second uniform lamina \(A B C D E F A\) is shown shaded in Figure 3. The straight sides \(A C\) and \(A E\) are perpendicular and \(A C = A E = 2 a\). In the figure, the midpoint of \(A C\) is \(B\), the midpoint of \(A E\) is \(F\), and \(A B D F\) and \(D G E F\) are squares of side \(a\). \(B C D\) is a quadrant of a circle with centre \(B\). \(D G E\) is a quadrant of a circle with centre \(G\).
2. Find the distance of the centre of mass of the lamina from the side \(A E\). The lamina is smoothly hinged to a horizontal axis which passes through \(E\) and is perpendicular to the plane of the lamina. The lamina has weight \(W\) newtons. The lamina is held in equilibrium in a vertical plane, with \(A\) vertically above \(E\), by a horizontal force of magnitude \(X\) newtons applied at \(C\).
3. Find \(X\) in terms of \(W\).
   

5. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = ( x + 1 ) ^ { 2 }\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 3. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).

1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).
   (8) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-08_558_492_1263_703} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} A uniform solid hemisphere is fixed to \(S\) to form a solid \(T\). The hemisphere has the same radius as the smaller plane face of \(S\) and its plane face coincides with the smaller plane face of \(S\), as shown in Figure 4. The mass per unit volume of the hemisphere is 10 times the mass per unit volume of \(S\). The centre of the circular plane face of \(T\) is \(A\). All lengths are measured in centimetres.
2. Find the distance of the centre of mass of \(T\) from \(A\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a container in the shape of a uniform right circular conical shell of height 6r. The radius of the open circular face is \(r\). The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is \(2 r\). The container together with the wax forms a solid \(S\). The mass of the container when empty is \(m\) and the mass of the wax in the container is \(3 m\).

1. Find the distance of the centre of mass of the solid \(S\) from the vertex of the container. One of the strings is now removed and the solid \(S\) hangs freely in equilibrium suspended by the remaining vertical string.
2. Find the size of the angle between the axis of the container and the downward vertical.

5. \begin{figure}[h] \end{figure} Figure 2 shows a uniform solid spindle which is made by joining together the circular faces of two right circular cones. The common circular face has radius \(r\) and centre \(O\). The smaller cone has height \(h\) and the larger cone has height \(k h\). The point \(A\) lies on the circumference of the common circular face. The spindle is suspended from \(A\) and hangs freely in equilibrium with \(A O\) at an angle of \(30 ^ { \circ }\) to the vertical. Show that \(k = \frac { 4 r } { h \sqrt { 3 } } + 1\)

3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\). Find

1. the distance \(O A\),
2. the magnitude of the instantaneous acceleration of \(P\) at \(A\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
   \end{figure} A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
   1. Show that the distance of the centre of mass of \(S\) from \(O\) is $$\frac { h } { 4 } ( k - 2 )$$ The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. Given that \(r = 3 h\) and \(k = 6\)
   2. find the size of the angle between \(A O\) and the vertical.

4. A uniform solid \(S\) consists of two right circular cones of base rate
has height \(2 h\) and the centre of the plane face of this cone is \(O\). T \(k h\) where \(k > 2\). The two cones are joined so that their plane fac
Figure 2 .

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { h } { 4 } ( k - 2 )\)

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(a\) and height \(4 a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(a\), is made of the same material as the cylinder. The hemispherical shell is attached to the open end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Find the distance from \(O\) to the centre of mass of \(C\). The container is placed with its circular base on a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(\theta\).

5. A uniform solid hemisphere has radius \(r\). The centre of the plane face of the hemisphere is \(O\).

1. Use algebraic integration to show that the distance from \(O\) to the centre of mass of the hemisphere is \(\frac { 3 } { 8 } r\).
   [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-14_378_602_740_671} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} A solid \(S\) is formed by joining a uniform solid hemisphere of radius \(a\) to a uniform solid hemisphere of radius \(\frac { 1 } { 2 } a\). The plane faces of the hemispheres are joined together so that their centres coincide at \(O\), as shown in Figure 1. The mass per unit volume of the smaller hemisphere is \(k\) times the mass per unit volume of the larger hemisphere.
2. Find the distance from \(O\) to the centre of mass of \(S\). When \(S\) is placed on a horizontal plane with any point on the curved surface of the larger hemisphere in contact with the plane, \(S\) remains in equilibrium.
3. Find the value of \(k\).

6.

1. Show, by integration, that the centre of mass of a uniform right cone, of radius \(a\) and height \(h\), is a distance \(\frac { 3 } { 4 } h\) from the vertex of the cone. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-5_789_914_406_486} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} A uniform right cone \(C\), of radius \(a\) and height \(h\), has vertex \(A\). A solid \(S\) is formed by removing from \(C\) another cone, of radius \(\frac { 2 } { 3 } a\) and height \(\frac { 1 } { 2 } h\), with the same axis as \(C\). The plane faces of the two cones coincide, as shown in Fig. 3.
2. Find the distance of the centre of mass of \(S\) from \(A\).
   

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

2. \begin{figure}[h] \end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
2. Find the value of \(k\).

A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.

A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.

2 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696} A uniform circular disc with centre \(A\) has mass \(M\) and radius \(3 a\). A second uniform circular disc with centre \(B\) has mass \(\frac { 1 } { 9 } M\) and radius \(a\). The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at \(P\) and the circumference of the smaller disc at \(O\). A particle of mass \(\frac { 1 } { 3 } M\) is attached at \(P\) (see diagram). Show that the moment of inertia of the system about an axis through \(O\), perpendicular to the plane of the discs, is \(51 M a ^ { 2 }\). The system is free to rotate about a fixed horizontal axis through \(O\), perpendicular to the plane of the discs. The system is held with \(O P\) horizontal and is then released from rest. Given that \(a = 0.5 \mathrm {~m}\), find the greatest speed of \(P\) in the subsequent motion, giving your answer correct to 2 significant figures.
[0pt] [5]

5 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598} The end \(B\) of a uniform rod \(A B\), of mass \(3 M\) and length \(4 a\), is rigidly attached to a point on the circumference of a uniform disc. The disc has centre \(O\), mass \(2 M\) and radius \(a\), and \(A B O\) is a straight line. The disc and the rod are in the same vertical plane. A particle \(P\), of mass \(M\), is attached to the rod at a distance \(k a\) from \(A\), where \(k\) is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis \(l\) through \(A\) perpendicular to the plane of the disc, is \(\left( 67 + k ^ { 2 } \right) M a ^ { 2 }\). The system is free to rotate about \(l\) and performs small oscillations of period \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\). Find the possible values of \(k\).

5 \includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.

1 \includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_216_1205_253_470} A rigid body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), the centres of which are rigidly attached to the ends \(A\) and \(B\) of a uniform rod of mass \(3 m\) and length \(10 a\). The discs and the rod are in the same plane and \(O\) is the point on the rod such that \(A O = 4 a\) (see diagram). Show that the moment of inertia of the body about an axis through \(O\) perpendicular to the plane of the discs is \(81 m a ^ { 2 }\).

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 \(A D\).