6.04c Composite bodies: centre of mass

1 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_533_497_269_824} A frame consists of a uniform circular ring of radius 25 cm and mass 1.5 kg , and a uniform rod of length 48 cm and mass 0.6 kg . The ends \(A\) and \(B\) of the rod are attached to points on the circumference of the ring, as shown in the diagram. Find the distance of the centre of mass of the frame from the centre of the ring.

2 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_291_732_822_708} A uniform lamina \(A B C D E\) consists of a rectangular part with sides 5 cm and 10 cm , and a part in the form of a quarter of a circle of radius 5 cm , as shown in the diagram.

1. Show that the distance of the centre of mass of the part \(C D E\) of the lamina is \(\frac { 20 } { 3 \pi } \mathrm {~cm}\) from \(C E\).
2. Find the distance of the centre of mass of the lamina \(A B C D E\) from the edge \(A B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-4_620_899_644_623} A rigid rod consists of two parts. The part \(B C\) is in the form of an arc of a circle of radius 2 m and centre \(O\), with angle \(B O C = \frac { 1 } { 4 } \pi\) radians. \(B C\) is uniform and has weight 3 N . The part \(A B\) is straight and of length 2 m ; it is uniform and has weight 4 N . The part \(A B\) of the rod is a tangent to the arc \(B C\) at \(B\). The end \(A\) of the rod is freely hinged to a fixed point of a vertical wall. The rod is held in equilibrium, with the straight part \(A B\) making an angle of \(\frac { 1 } { 4 } \pi\) radians with the wall, by means of a horizontal string attached to \(C\). The string is in the same vertical plane as the rod, and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Show that the centre of mass \(G\) of the part \(B C\) of the rod is at a distance of 2.083 m from the wall, correct to 4 significant figures.
2. Find the value of \(T\).
3. State the magnitude of the horizontal component and the magnitude of the vertical component of the force exerted on the rod by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-5_579_1118_264_516} A particle \(A\) is released from rest at time \(t = 0\), at a point \(P\) which is 7 m above horizontal ground. At the same instant as \(A\) is released, a particle \(B\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(P\) is 24 m . Particle \(B\) moves in the vertical plane containing \(O\) and \(P\), with initial speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle of \(\theta\) above the horizontal (see diagram). Write down
4. an expression for the height of \(A\) above the ground at time \(t \mathrm {~s}\),
5. an expression in terms of \(V , \theta\) and \(t\) for
   1. the horizontal distance of \(B\) from \(O\),
   2. the height of \(B\) above the ground. At time \(t = T\) the particles \(A\) and \(B\) collide at a point above the ground.
   3. Show that \(\tan \theta = \frac { 7 } { 24 }\) and that \(V T = 25\).
   4. Deduce that \(7 V ^ { 2 } > 3125\).

4 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-3_777_447_267_849} A uniform triangular lamina \(A B C\) is right-angled at \(B\) and has sides \(A B = 0.6 \mathrm {~m}\) and \(B C = 0.8 \mathrm {~m}\). The mass of the lamina is 4 kg . One end of a light inextensible rope is attached to the lamina at \(C\). The other end of the rope is attached to a fixed point \(D\) on a vertical wall. The lamina is in equilibrium with \(A\) in contact with the wall at a point vertically below \(D\). The lamina is in a vertical plane perpendicular to the wall, and \(A B\) is horizontal. The rope is taut and at right angles to \(A C\) (see diagram). Find

1. the tension in the rope,
2. the horizontal and vertical components of the force exerted at \(A\) on the lamina by the wall.

2 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_686_495_1238_826} A uniform rigid wire \(A B\) is in the form of a circular arc of radius 1.5 m with centre \(O\). The angle \(A O B\) is a right angle. The wire is in equilibrium, freely suspended from the end \(A\). The chord \(A B\) makes an angle of \(\theta ^ { \circ }\) with the vertical (see diagram).

1. Show that the distance of the centre of mass of the arc from \(O\) is 1.35 m , correct to 3 significant figures.
2. Find the value of \(\theta\).

4 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_375_627_1448_758} Uniform rods \(A B , A C\) and \(B C\) have lengths \(3 \mathrm {~m} , 4 \mathrm {~m}\) and 5 m respectively, and weights \(15 \mathrm {~N} , 20 \mathrm {~N}\) and 25 N respectively. The rods are rigidly joined to form a right-angled triangular frame \(A B C\). The frame is hinged at \(B\) to a fixed point and is held in equilibrium, with \(A C\) horizontal, by means of an inextensible string attached at \(C\). The string is at right angles to \(B C\) and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Find the value of \(T\). A uniform triangular lamina \(P Q R\), of weight 60 N , has the same size and shape as the frame \(A B C\). The lamina is now attached to the frame with \(P , Q\) and \(R\) at \(A , B\) and \(C\) respectively. The composite body is held in equilibrium with \(A , B\) and \(C\) in the same positions as before. Find
2. the new value of \(T\),
3. the magnitude of the vertical component of the force acting on the composite body at \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-2_839_330_539_906} \(A B\) is a diameter of a uniform solid hemisphere with centre \(O\), radius 10 cm and weight 12 N . One end of a light inextensible string is attached to the hemisphere at \(B\) and the other end is attached to a fixed point \(C\) of a vertical wall. The hemisphere is in equilibrium with \(A\) in contact with the wall at a point vertically below \(C\). The centre of mass \(G\) of the hemisphere is at the same horizontal level as \(A\), and angle \(A B C\) is a right angle (see diagram). Calculate the tension in the string.

1 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-2_618_441_253_852} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg , and a uniform straight wire of length 40 cm and mass 0.9 kg . The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire.

4 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_727_565_1256_790} A uniform lamina of weight 15 N is in the form of a trapezium \(A B C D\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(A B\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30 ^ { \circ }\) upwards from the horizontal (see diagram). Find the tension in the string.

2 A uniform lamina \(A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and diameter 0.4 m , and an isosceles triangle \(A B D\) with base \(B D = 0.4 \mathrm {~m}\) and perpendicular height \(h \mathrm {~m}\). The centre of mass of the lamina is at \(O\).

1. Find the value of \(h\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_680_627_1466_797} The lamina is suspended from a vertical string attached to a point \(X\) on the side \(A D\) of the triangle (see diagram). Given the lamina is in equilibrium with \(A D\) horizontal, calculate \(X D\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-4_597_1011_251_566} \(A B C D E\) is the cross-section through the centre of mass of a uniform prism resting in equilibrium with \(D E\) on a horizontal surface. The cross-section has the shape of a square \(O B C D\) with sides of length \(a \mathrm {~m}\), from which a quadrant \(O A E\) of a circle of radius 1 m has been removed (see diagram).

1. Find the distance of the centre of mass of the prism from \(O\), giving the answer in terms of \(a , \pi\) and \(\sqrt { } 2\).
2. Hence show that $$3 a ^ { 2 } ( 2 - a ) < \frac { 3 } { 2 } \pi - 2$$ and verify that this inequality is satisfied by \(a = 1.68\) but not by \(a = 1.67\).

2 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_481_412_440_865} The diagram shows a circular object formed from a uniform semicircular lamina of weight 11 N and a uniform semicircular arc of weight 9 N . The lamina and the arc both have centre \(O\) and radius 0.7 m and are joined at the ends of their common diameter \(A B\).

1. Show that the distance of the centre of mass of the object from \(O\) is 0.0371 m , correct to 3 significant figures. The object hangs in equilibrium, freely suspended at \(A\).
2. Find the angle between \(A B\) and the vertical and state whether the lowest point of the object is on the lamina or on the arc.

6 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-3_720_723_1165_712} The diagram shows the cross-section \(O A B C D E\) through the centre of mass of a uniform prism. The interior angles of the cross-section at \(O , A , C , D\) and \(E\) are all right angles. \(O A = 0.4 \mathrm {~m} , A B = 0.5 \mathrm {~m}\) and \(B C = C D = 1 \mathrm {~m}\).

1. Calculate the distance of the centre of mass of the prism from \(O E\). The weight of the prism is 120 N . A force of magnitude \(F \mathrm {~N}\) acting along \(D E\) holds the prism in equilibrium when \(O A\) rests on a rough horizontal surface.
2. Find the set of possible values of \(F\).

2 A uniform hemispherical shell of weight 8 N and a uniform solid hemisphere of weight 12 N are joined along their circumferences to form a non-uniform sphere of radius 0.2 m .

1. Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m . This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude \(F \mathrm {~N}\) acting parallel to the axis of symmetry applied to the highest point of the sphere.
2. Calculate \(F\).

6 \includegraphics[max width=\textwidth, alt={}, center]{09971be0-73b6-4c73-8dfd-c89ff877950a-3_451_775_255_685} The diagram shows a uniform lamina \(A B C D E F\), formed from a semicircle with centre \(O\) and radius 1 m by removing a semicircular part with centre \(O\) and radius \(r \mathrm {~m}\).

1. Show that the distance in metres of the centre of mass of the lamina from \(O\) is $$\frac { 4 \left( 1 + r + r ^ { 2 } \right) } { 3 \pi ( 1 + r ) } .$$ The centre of mass of the lamina lies on the \(\operatorname { arc } A B C\).
2. Show that \(r = 0.494\), correct to 3 significant figures. The lamina is freely suspended at \(F\) and hangs in equilibrium.
3. Find the angle between the diameter of the lamina and the vertical.

6 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-3_341_791_886_678} A uniform lamina \(A B C D E\) consists of a rectangle \(B C D E\) and an isosceles triangle \(A B E\) joined along their common edge \(B E\). For the triangle, \(A B = A E , B E = a \mathrm {~m}\) and the perpendicular height is \(h \mathrm {~m}\). For the rectangle, \(B C = D E = 0.5 \mathrm {~m}\) and \(C D = B E = a \mathrm {~m}\) (see diagram).

1. Show that the distance in metres of the centre of mass of the lamina from \(B E\) towards \(C D\) is $$\frac { 3 - 4 h ^ { 2 } } { 12 + 12 h }$$ The lamina is freely suspended at \(E\) and hangs in equilibrium.
2. Given that \(D E\) is horizontal, calculate \(h\).
3. Given instead that \(h = 0.5\) and \(A E\) is horizontal, calculate \(a\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-2_373_759_1119_694} A uniform object \(A B C\) is formed from two rods \(A B\) and \(B C\) joined rigidly at right angles at \(B\). The rod \(A B\) has length 0.3 m and the rod \(B C\) has length 0.2 m . The object rests with the end \(A\) on a rough horizontal surface and the \(\operatorname { rod } A B\) vertical. The object is held in equilibrium by a horizontal force of magnitude 4 N applied at \(B\) and acting in the direction \(C B\) (see diagram).

1. Find the distance of the centre of mass of the object from \(A B\).
2. Calculate the weight of the object.
3. Find the least possible value of the coefficient of friction between the surface and the object.

2 A uniform semicircular lamina of radius 0.25 m has diameter \(A B\). It is freely suspended at \(A\) from a fixed point and hangs in equilibrium.

1. Find the distance of the centre of mass of the lamina from the diameter \(A B\).
2. Calculate the angle which the diameter \(A B\) makes with the vertical. The lamina is now held in equilibrium with the diameter \(A B\) vertical by means of a force applied at \(B\). This force has magnitude 6 N and acts at \(45 ^ { \circ }\) to the upward vertical in the plane of the lamina.
3. Calculate the weight of the lamina.

6 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_291_993_1238_575} A uniform solid cone of height 0.6 m and mass 0.5 kg has its axis of symmetry vertical and its vertex \(V\) uppermost. The semi-vertical angle of the cone is \(60 ^ { \circ }\) and the surface is smooth. The cone is fixed to a horizontal surface. A particle \(P\) of mass 0.2 kg is connected to \(V\) by a light inextensible string of length 0.4 m (see diagram).

1. Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle. \(P\) is set in motion, and moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the cone.
2. Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on \(P\) by the cone.
3. Find the speed of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-3_474_860_1288_644} A uniform solid cone of height 1.2 m and semi-vertical angle \(\theta ^ { \circ }\) is divided into two parts by a cut parallel to and 0.4 m from the circular base. The upper conical part, \(C\), has weight 16 N , and the lower part, \(L\), has weight 38 N . The two parts of the solid rest in equilibrium with the larger plane face of \(L\) on a horizontal surface and the smaller plane face of \(L\) covered by the base of \(C\) (see diagram).

1. Calculate the distance of the centre of mass of \(L\) from its larger plane face. An increasing horizontal force is applied to the vertex of \(C\). Equilibrium is broken when the magnitude of this force first exceeds 4 N , and \(C\) begins to slide on \(L\).
2. By considering the forces on \(C\),
   (a) find the coefficient of friction between \(C\) and \(L\),
   (b) show that \(\theta > 14.0\), correct to 3 significant figures. \(C\) is removed and \(L\) is placed with its curved surface on the horizontal surface.
3. Given that \(L\) is on the point of toppling, calculate \(\theta\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-4_553_630_258_753} The diagram shows a container which consists of a bowl of weight 14 N and a handle of weight 8 N . The bowl of the container is in the form of a uniform hemispherical shell with centre \(O\) and radius 0.3 m . The handle is in the form of a uniform semicircular arc of radius 0.3 m and is freely hinged to the bowl at \(A\) and \(B\), where \(A B\) is a diameter of the bowl.

1. Calculate the distance of the centre of mass of the container from \(O\) for the position indicated in the diagram, where the handle is perpendicular to the rim of the bowl.
2. Show that the distance of the centre of mass of the container from \(O\) when the handle lies on the rim of the bowl is 0.118 m , correct to 3 significant figures. In the case when the handle lies on the rim of the bowl, the container rests in equilibrium with the curved surface of the bowl on a horizontal table.
3. Find the angle which the plane containing the rim of the bowl makes with the horizontal.

3 A triangular frame \(A B C\) consists of two uniform rigid rods each of length 0.8 m and weight 3 N , and a longer uniform rod of weight 4 N . The triangular frame has \(A B = B C\), and angle \(B A C =\) angle \(B C A = 30 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the frame from \(A C\). \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-2_722_335_1302_904} The vertex \(A\) of the frame is attached to a smooth hinge at a fixed point. The frame is held in equilibrium with \(A C\) vertical by a vertical force of magnitude \(F \mathrm {~N}\) applied to the frame at \(B\) (see diagram).
2. Calculate \(F\), and state the magnitude and direction of the force acting on the frame at the hinge.

1 A uniform semicircular lamina has diameter \(A B\) of length 0.8 m .

1. Find the distance of the centre of mass of the lamina from \(A B\). The lamina rests in a vertical plane, with the point \(B\) of the lamina in contact with a rough horizontal surface and with \(A\) vertically above \(B\). Equilibrium is maintained by a force of magnitude 6 N in the plane of the lamina, applied to the lamina at \(A\) and acting at an angle of \(20 ^ { \circ }\) below the horizontal.
2. Calculate the mass of the lamina.

5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).

1. Show that the tension in the string is 60 N at the instant when the prism topples.
2. Calculate the speed of the particle at the instant when the prism topples.

4 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-3_784_556_260_790} A uniform object is made by drilling a cylindrical hole through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section \(A B C D\) through the centre of mass of the object. \(A B = C D = 0.7 \mathrm {~m} , B C = A D = 0.4 \mathrm {~m}\), and the centre of the hole is 0.1 m from \(A B\) and 0.2 m from \(A D\) (see diagram). The hole has a cross-section of area \(0.03 \mathrm {~m} ^ { 2 }\).

1. Show that the distance of the centre of mass of the object from \(A B\) is 0.212 m , and calculate the distance of the centre of mass from \(A D\). The object has weight 70 N and is placed on a rough horizontal surface, with \(A D\) in contact with the surface. A vertically upwards force of magnitude \(F \mathrm {~N}\) acts on the object at \(C\). The object is on the point of toppling.
2. Find the value of \(F\). The force acting at \(C\) is removed, and the object is placed on a rough plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal. \(A D\) lies along a line of greatest slope, with \(A\) higher than \(D\). The plane is sufficiently rough to prevent sliding, and the object does not topple.
3. Find the greatest possible value of \(\theta\).