6.04c Composite bodies: centre of mass

3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .

1. Calculate the height of the centre of mass of the box above its base.
   The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
2. Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.

7 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
   [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
2. Find the distance of the centre of mass of the object from its base.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-04_442_554_260_794} A uniform lamina \(A B C E F G\) is formed from a square \(A B D G\) by removing a smaller square \(C D F E\) from one corner. \(A B = 0.7 \mathrm {~m}\) and \(D F = 0.3 \mathrm {~m}\) (see diagram). Find the distance of the centre of mass of the lamina from \(A\).

4 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_549_579_1505_781} A uniform lamina is made by joining a rectangle \(A B C D\), in which \(A B = C D = 0.56 \mathrm {~m}\) and \(B C = A D = 2 \mathrm {~m}\), and a square \(E F G A\) of side 1.2 m . The vertex \(E\) of the square lies on the edge \(A D\) of the rectangle (see diagram). The centre of mass of the lamina is a distance \(h \mathrm {~m}\) from \(B C\) and a distance \(v \mathrm {~m}\) from \(B A G\).

1. Find the value of \(h\) and show that \(v = h\). The lamina is freely suspended at the point \(B\) and hangs in equilibrium.
2. State the angle which the edge \(B C\) makes with the horizontal. Instead, the lamina is now freely suspended at the point \(F\) and hangs in equilibrium.
3. Calculate the angle between \(F G\) and the vertical.

2 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-04_606_376_260_881} A uniform object is made by joining together three solid cubes with edges \(3 \mathrm {~m} , 2 \mathrm {~m}\) and 1 m . The object has an axis of symmetry, with the cubes stacked vertically and the cube of edge 2 m between the other two cubes (see diagram).

1. Calculate the distance of the centre of mass of the object above the base of the largest cube.
   The smallest cube is now removed from the object. It is replaced by a heavier uniform cube with 1 m edges which is made of a different material. The centre of mass of the object is now at the base of the 2 m cube.
2. Find the ratio of the masses of the two cubes of edge 1 m .

7 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).

1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
2. Find the value of \(y\) for which the container is about to topple.

3 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_293_1045_267_550} A uniform lamina \(A B C D\) is in the form of a trapezium in which \(A B\) and \(D C\) are parallel and have lengths 2 m and 3 m respectively. \(B D\) is perpendicular to the parallel sides and has length 1 m (see diagram).

1. Find the distance of the centre of mass of the lamina from \(B D\). The lamina has weight \(W \mathrm {~N}\) and is in equilibrium, suspended by a vertical string attached to the lamina at \(B\). The lamina rests on a vertical support at \(C\). The lamina is in a vertical plane with \(A B\) and \(D C\) horizontal.
2. Find, in terms of \(W\), the tension in the string and the magnitude of the force exerted on the lamina at \(C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_597_690_1416_731} A large uniform lamina is in the shape of a right-angled triangle \(A B C\), with hypotenuse \(A C\), joined to a semicircle \(A D C\) with diameter \(A C\). The sides \(A B\) and \(B C\) have lengths 3 m and 4 m respectively, as shown in the diagram.

1. Show that the distance from \(A B\) of the centre of mass of the semicircular part \(A D C\) of the lamina is \(\left( 2 + \frac { 2 } { \pi } \right) \mathrm { m }\).
2. Show that the distance from \(A B\) of the centre of mass of the complete lamina is 2.14 m , correct to 3 significant figures.

4 A particle is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 0.3 s the particle is moving with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\tan ^ { - 1 } \left( \frac { 7 } { 24 } \right)\) above the horizontal.

1. Show that \(V \cos \theta = 24\).
2. Find the value of \(V \sin \theta\), and hence find \(V\) and \(\theta\).

3 An object consists of a uniform solid circular cone, of vertical height \(4 r\) and radius \(3 r\), and a uniform solid cylinder, of height \(4 r\) and radius \(3 r\). The circular base of the cone and one of the circular faces of the cylinder are joined together so that they coincide. The cone and the cylinder are made of the same material.

1. Find the distance of the centre of mass of the object from the end of the cylinder that is not attached to the cone.
2. Show that the object can rest in equilibrium with the curved surface of the cone in contact with a horizontal surface.

3 One end of a light elastic spring, of natural length \(a\) and modulus of elasticity 5 mg , is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac { 1 } { 2 } a\).

1. Show that the initial acceleration of \(P\) is \(\frac { 3 } { 2 } g\) upwards.
2. Find the speed of \(P\) when the spring first returns to its natural length. \includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-08_581_659_267_708} A uniform square lamina \(A B C D\) has sides of length 10 cm . The point \(E\) is on \(B C\) with \(E C = 7.5 \mathrm {~cm}\), and the point \(F\) is on \(D C\) with \(\mathrm { CF } = \mathrm { xcm }\). The triangle \(E F C\) is removed from \(A B C D\) (see diagram). The centre of mass of the resulting shape \(A B E F D\) is a distance \(\bar { x } \mathrm {~cm}\) from \(C B\) and a distance \(\bar { y } \mathrm {~cm}\) from CD.

1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\). Find the \(x\)-coordinate of the centre of mass of the lamina.

3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

2 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-2_673_401_525_872} A bow consists of a uniform curved portion \(A B\) of mass 1.4 kg , and a uniform taut string of mass \(m \mathrm {~kg}\) which joins \(A\) and \(B\). The curved portion \(A B\) is an arc of a circle centre \(O\) and radius 0.8 m . Angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m from \(O\). Calculate \(m\).

2 An object is made from two identical uniform rods \(A B\) and \(B C\) each of length 0.6 m and weight 7 N . The rods are rigidly joined to each other at \(B\) and angle \(A B C = 90 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the object from \(B\). The object is freely suspended at \(A\) and a force of magnitude \(F \mathrm {~N}\) is applied to the rod \(B C\) at \(C\). The object is in equilibrium with \(A B\) inclined at \(45 ^ { \circ }\) to the horizontal.
2. (a) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_401_314_799_995} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Calculate \(F\) given that the force acts horizontally as shown in Fig. 1.
   (b) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_503_273_1446_1014} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Calculate \(F\) given instead that the force acts perpendicular to the rod as shown in Fig. 2.

6 A uniform solid consists of a hemisphere with centre \(O\) and radius 0.6 m joined to a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere coincides with one of the plane faces of the cylinder.

1. Calculate the distance of the centre of mass of the solid from \(O\).
   [0pt] [The volume of a hemisphere of radius \(r\) is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
2. \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-4_547_631_593_797} A cylindrical hole, of length 0.48 m , starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at \(O\). Show that the area of the cross-section of the hole is \(\frac { 3 } { 16 } \pi \mathrm {~m} ^ { 2 }\).
3. It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at \(O\). State the increase in the length of the hole.

6 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-3_582_862_577_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).

1. Show that the centre of mass of the lamina lies on \(O B\).
2. Calculate the distance of the centre of mass of the lamina from \(O\).

1 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_426_531_258_808} A circular object is formed from a uniform semicircular lamina of weight 12 N and a uniform semicircular arc of weight 8 N . The lamina and the arc both have centre \(O\) and radius 0.6 m and are joined at the ends of their common diameter \(A B\). The object is freely pivoted to a fixed point at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. The object is in equilibrium acted on by a horizontal force of magnitude \(F\) N applied at the lowest point of the object, and acting in the plane of the object (see diagram).

1. Show that the centre of mass of the object is at \(O\).
2. Calculate \(F\).

2 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-2_359_686_484_731} A uniform frame consists of a semicircular arc \(A B C\) of radius 0.6 m together with its diameter \(A O C\), where \(O\) is the centre of the semicircle (see diagram).

1. Calculate the distance of the centre of mass of the frame from \(O\). The frame is freely suspended at \(A\) and hangs in equilibrium.
2. Calculate the angle between \(A C\) and the vertical.

6 \includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-3_424_953_255_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.

1. Show that \(d = h + \frac { 0.04 } { h }\).
2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).

2 \includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-2_318_495_484_824} A uniform wire is bent to form an object which has a semicircular arc with diameter \(A B\) of length 1.2 m , with a smaller semicircular arc with diameter \(B C\) of length 0.6 m . The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

1. Show that the distance of the centre of mass of the object from the line \(A C B\) is 0.191 m , correct to 3 significant figures. The object is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between \(A C B\) and the vertical.

4 \includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_388_650_264_749} The diagram shows the cross-section \(A B C D\) through the centre of mass of a uniform solid prism. \(A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}\) and angle \(A B C =\) angle \(B A D = 90 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the prism from \(A D\).
2. Express the distance of the centre of mass of the prism from \(A B\) in terms of \(a\). The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(A D\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(B C\).
3. Given that the prism is on the point of toppling, calculate \(a\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-3_656_757_781_694} The diagram shows the cross-section \(A B C D E F\) through the centre of mass of a uniform prism which rests with \(A B\) on rough horizontal ground. \(A B C D\) is a rectangle with \(A B = C D = 0.4 \mathrm {~m}\) and \(B C = A D = 1.8 \mathrm {~m}\). The other part of the cross-section is a semicircle with diameter \(D F\) and radius \(r \mathrm {~m}\).

1. Given that the prism is on the point of toppling, show that \(r = 0.6\). A force of magnitude \(P \mathrm {~N}\) is applied to the prism, acting at \(60 ^ { \circ }\) to the upwards vertical along a tangent to the semicircle at a point between \(D\) and \(E\). The prism has weight 15 N and is in equilibrium on the point of toppling about \(B\).
2. Show that \(P = 3.26\), correct to 3 significant figures.
3. Find the smallest possible value of the coefficient of friction between the prism and the ground.

6 A solid object consists of a uniform hemisphere of radius 0.4 m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N . The centre of mass of the object lies at the centre \(O\) of their common circular face.

1. Show that the height of the cylinder is 0.3 m .
   A new object is made by cutting the cylinder in half and removing the half not attached to the hemisphere. The cut is perpendicular to the axis of symmetry, so the new object consists of a hemisphere and a cylinder half the height of the original cylinder.
2. Find the distance of the centre of mass of the new object from \(O\).
   The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting along its axis of symmetry, which is inclined at \(30 ^ { \circ }\) to the horizontal.
3. Find \(P\).

2 A uniform solid cone has height 0.6 m and base radius 0.2 m . A uniform hollow cylinder, open at both ends, has the same dimensions. An object is made by putting the cone inside the cylinder so that the base of the cone coincides with one end of the cylinder (see diagram, which shows a cross-section). The total weight of the object is 60 N and its centre of mass is 0.25 m from the base of the cone. Calculate the weight of the cone.