6.04d Integration: for centre of mass of laminas/solids

2. \begin{figure}[h] \end{figure} A uniform circular disc has mass \(4 m\), centre \(O\) and radius \(4 a\). The line \(P O Q\) is a diameter of the disc. A circular hole of radius \(2 a\) is made in the disc with the centre of the hole at the point \(R\) on \(P Q\) where \(Q R = 5 a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.

1. Show that the moment of inertia of the lamina about \(L\) is \(69 m a ^ { 2 }\). The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac { 2 \pi } { 3 }\) before it first comes to instantaneous rest,
2. find \(\Omega\) in terms of \(g\) and \(a\).

A uniform lamina \(A B C\) of mass \(m\) is in the shape of an isosceles triangle with \(A B = A C = 5 a\) and \(B C = 8 a\).

1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(B C\), is \(\frac { 9 } { 2 } m a ^ { 2 }\). The foot of the perpendicular from \(A\) to \(B C\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(D A\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(A\), perpendicular to \(B C\) and in the plane of the lamina, is \(\frac { 8 } { 3 } m a ^ { 2 }\).
2. Find the angular acceleration of the lamina when \(D A\) makes an angle \(\theta\) with the downward vertical. Given that \(\alpha\) is small,
3. find an approximate value for the period of oscillation of the lamina about the vertical.

4. A uniform lamina of mass \(M \mathrm {~kg}\) is modelled as the region which is bounded by the curve with equation \(y = x ^ { 2 }\), the positive \(x\)-axis and the line with equation \(x = 2\). The unit of length on both axes is the metre. Find the moment of inertia of the lamina about the \(x\)-axis.
(6)

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

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 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.

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 .

4 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina. Determine the exact \(x\)-coordinate of the centre of mass of the lamina.

5.

1. 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.
2. 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.
   

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 \(\tan \alpha\).
   

4. The diagram shows a uniform lamina consisting of a rectangular section \(G P Q E\) with a semi-circular section EFG of radius 4 cm . Quadrants \(A P B\) and \(C Q D\) each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

1. Write down the distance of the centre of mass of the lamina \(A B C D E F G\) from \(A G\).
2. Determine the distance of the centre of mass of the lamina \(A B C D E F G\) from \(B C\).
3. The lamina \(A B C D E F G\) is suspended freely from the point \(E\) and hangs in equilibrium. Calculate the angle EG makes with the vertical.

6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6

1. Find the centre of mass of this solid.
   6
2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)

1. Numerical (calculator) integration is not acceptable in this question.

\begin{figure}[h] \end{figure} The shaded region \(O A B\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac { 1 } { 4 } ( x - 2 ) ^ { 3 } + 2\). The point \(A\) has coordinates (4, 4) and the point \(B\) has coordinates \(( 4,0 )\). A uniform lamina \(L\) has the shape of \(O A B\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \(( \bar { x } , \bar { y } )\). Given that the area of \(L\) is \(8 \mathrm {~cm} ^ { 2 }\),

1. show that \(\bar { y } = \frac { 8 } { 7 }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\).

5. \begin{figure}[h] \end{figure} The region \(R\), shown shaded in Figure 4, is bounded by part of the curve with equation \(y ^ { 2 } = 2 x\), the line with equation \(y = 2\) and the \(y\)-axis. The unit of length on both axes is one centimetre. A uniform solid, \(S\), is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
Given that the volume of \(S\) is \(\frac { 8 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),

1. show that the centre of mass of \(S\) is \(\frac { 1 } { 3 } \mathrm {~cm}\) from its plane face. A uniform solid cylinder, \(C\), has base radius 2 cm and height 4 cm . The cylinder \(C\) is attached to \(S\) so that the plane face of \(S\) coincides with a plane face of \(C\), to form the paperweight \(P\), shown in Figure 5. The density of the material used to make \(S\) is three times the density of the material used to make \(C\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_572_456_1617_758} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The plane face of \(P\) rests in equilibrium on a desk lid that is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The lid is sufficiently rough to prevent \(P\) from slipping. Given that \(P\) is on the point of toppling,
2. find the value of \(\theta\).

2. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A uniform plane figure \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = \ln 5\), the curve with equation \(y = 8 \mathrm { e } ^ { - x }\) and the line with equation \(x = \ln 2\). The unit of length on each axis is one metre. The area of \(R\) is \(2.4 \mathrm {~m} ^ { 2 }\) The centre of mass of \(R\) is at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. Use algebraic integration to show that \(\bar { y } = 1.4\) Figure 2 shows a uniform lamina \(A B C D\), which is the same size and shape as \(R\). The lamina is freely suspended from \(C\) and hangs in equilibrium with \(C B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\)

1. \hspace{0pt} [In this question, you may assume that the centre of mass of a circular arc, radius \(r\), with angle at centre \(2 \alpha\), is a distance \(\frac { r \sin \alpha } { \alpha }\) from the centre.]

\begin{figure}[h] \end{figure} A thin non-uniform metal plate is in the shape of a sector \(O A B\) of a circle with centre \(O\) and radius \(a\). The angle \(A O B = \frac { \pi } { 2 }\), as shown in Figure 5. The plate is modelled as a non-uniform lamina.
The mass per unit area of the lamina, at any point \(P\) of the lamina, is modelled as \(k ( O P ) ^ { 2 }\), where \(k = \frac { 4 \lambda } { \pi a ^ { 4 } }\) and \(\lambda\) is a constant. Using the model,

1. find the mass of the plate in terms of \(\lambda\),
2. find, in terms of \(a\), the distance of the centre of mass of the plate from \(O\).

3. \begin{figure}[h] \end{figure} Nine uniform rods are joined together to form the rigid framework \(A B C D E F A\), with \(A B = B C = D F = 3 a , B F = C D = D E = 4 a\) and \(A F = F E = C F = 5 a\), as shown in Figure 1. All nine rods lie in the same plane. The mass per unit length of each of the rods \(B F , C F\) and \(D F\) is twice the mass per unit length of each of the other six rods.

1. Find the distance of the centre of mass of the framework from \(A C\) The mass of the framework is \(M\). A particle of mass \(k M\) is attached to the framework at \(E\) to form a loaded framework. When the loaded framework is freely suspended from \(F\), it hangs in equilibrium with \(C E\) horizontal.
2. Find the exact value of \(k\)

5. \begin{figure}[h] \end{figure} The uniform plane lamina shown in Figure 3 is formed from two squares, \(A B C O\) and \(O D E F\), and a sector \(O D C\) of a circle with centre \(O\). Both squares have sides of length \(3 a\) and \(A O\) is perpendicular to \(O F\). The radius of the sector is \(3 a\) [0pt] [In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.]

1. Show that the distance of the centre of mass of the sector \(O D C\) from \(O C\) is \(\frac { 4 a } { \pi }\)
2. Find the distance of the centre of mass of the lamina from \(F C\) The lamina is freely suspended from \(F\) and hangs in equilibrium with \(F C\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)