6.04e Rigid body equilibrium: coplanar forces

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641} \(O A B C\) is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre \(O\), radius \(O A = r \mathrm {~m}\) and angle \(A O C = \frac { 2 } { 3 } \pi\) radians. The prism lies on a plane inclined at an angle \(\theta\) radians to the horizontal, where \(\theta < \frac { 1 } { 3 } \pi\). OC lies along a line of greatest slope with \(O\) higher than \(C\). The prism is freely hinged to the plane at \(O\). A force of magnitude 15 N acts at \(A\), in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of \(\theta\).

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]{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 \(\theta\).

2 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-2_536_905_520_621} A uniform lamina \(A B C\) in the shape of an isosceles triangle has weight 24 N . The perpendicular distance from \(A\) to \(B C\) is 12 cm . The lamina rests in a vertical plane in equilibrium, with the vertex \(A\) in contact with a horizontal surface. Angle \(B A C = 100 ^ { \circ }\) and \(A B\) makes an angle of \(10 ^ { \circ }\) with the horizontal. Equilibrium is maintained by a force of magnitude \(F \mathrm {~N}\) acting along \(B C\) (see diagram). Show that \(F = 8\).

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-3_499_721_715_712} A uniform solid cube with edges of length 0.4 m rests in equilibrium on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A B C D\) is a cross-section through the centre of mass of the cube, with \(A B\) along a line of greatest slope. \(B\) lies below the level of \(A\). One end of a light elastic string with modulus of elasticity 12 N and natural length 0.4 m is attached to \(C\). The other end of the string is attached to a point below the level of \(B\) on the same line of greatest slope, such that the string makes an angle of \(30 ^ { \circ }\) with the plane (see diagram). The cube is on the point of toppling. Find

1. the tension in the string,
2. the weight of the cube.

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.

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 \(\theta\).

2 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671} A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of \(\theta\) to the horizontal, where \(\cos \theta = 0.28\). Equilibrium is maintained by a horizontal force of magnitude \(P\) N applied to the lowest point of the circular rim of the hemisphere (see diagram).

1. Show that \(P = 8.75\).
2. Find the coefficient of friction between the hemisphere and the plane.

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]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_319_874_968_639} A uniform rod \(A B\), of length 2 m and mass 10 kg , is freely hinged to a fixed point at the end \(B\). A light elastic string, of modulus of elasticity 200 N , has one end attached to the end \(A\) of the rod and the other end attached to a fixed point \(O\), which is in the same vertical plane as the rod. The rod is horizontal and in equilibrium, with \(O A = 3 \mathrm {~m}\) and angle \(O A B = 150 ^ { \circ }\) (see diagram). Find

1. the tension in the string,
2. the natural length of the string.

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A uniform solid cone has height 20 cm and base radius 10 cm . It is placed with its axis vertical on a rough horizontal plane (see Fig. 1). The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(\theta ^ { \circ }\), when the cone begins to topple without sliding (see Fig. 2).

1. Find the value of \(\theta\).
2. What can you say about the value of the coefficient of friction between the cone and the plane?

2 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-2_333_737_762_705} A uniform rod \(A B\) of length 1.2 m and weight 30 N is in equilibrium with the end \(A\) in contact with a vertical wall. \(A B\) is held at right angles to the wall by a light inextensible string. The string has one end attached to the rod at \(B\) and the other end attached to a point \(C\) of the wall. The point \(C\) is 0.5 m vertically above \(A\) (see diagram). Find

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

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_552_604_264_772} A uniform solid cone has vertical height 28 cm and base radius 6 cm . The cone is held with a point of the circumference of its base in contact with a horizontal table, and with the base making an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). When the cone is released, it moves towards the equilibrium position in which its base is in contact with the table. Show that \(\theta < 40.6\), correct to 1 decimal place.

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 \(C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_383_1031_543_557} A non-uniform rod \(A B\) of length 2.5 m and mass 3 kg has its centre of mass at the point \(G\) of the rod, where \(A G = 1.5 \mathrm {~m}\). The rod hangs horizontally, in equilibrium, from strings attached at \(A\) and \(B\). The strings at \(A\) and \(B\) make angles with the vertical of \(\alpha ^ { \circ }\) and \(15 ^ { \circ }\) respectively. The tension in the string at \(B\) is \(T \mathrm {~N}\) (see diagram). Find

1. the value of \(T\),
2. the value of \(\alpha\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).

1. Find the tension in the string. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally away from the wall and \(Y \mathrm {~N}\) vertically downwards.
2. Find the values of \(X\) and \(Y\).

6 A particle is projected from a point \(O\) at an angle of \(35 ^ { \circ }\) above the horizontal. At time \(T\) s later the particle passes through a point \(A\) whose horizontal and vertically upward displacements from \(O\) are 8 m and 3 m respectively.

1. By using the equation of the particle's trajectory, or otherwise, find (in either order) the speed of projection of the particle from \(O\) and the value of \(T\).
2. Find the angle between the direction of motion of the particle at \(A\) and the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_476_895_269_625} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross-section of a uniform solid. The cross-section has the shape and dimensions shown. The centre of mass \(C\) of the solid lies in the plane of this cross-section. The distance of \(C\) from \(D E\) is \(y \mathrm {~cm}\).
3. Find the value of \(y\). The solid is placed on a rough plane. The coefficient of friction between the solid and the plane is \(\mu\). The plane is tilted so that \(E F\) lies along a line of greatest slope.
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_375_431_1366_897} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The solid is placed so that \(F\) is higher up the plane than \(E\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 1 } { 2 }\). [3]
5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_376_428_2069_900} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The solid is now placed so that \(E\) is higher up the plane than \(F\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Show that \(\mu < \frac { 5 } { 6 }\). [3]

2 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-2_485_863_495_641} A uniform solid cylinder has height 24 cm and radius \(r \mathrm {~cm}\). A uniform solid cone has base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder and the cone are both placed with their axes vertical on a rough horizontal plane (see diagram, which shows cross-sections of the solids). The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha ^ { \circ }\), when both solids topple simultaneously.

1. Find the value of \(h\).
2. Given that \(r = 10\), find the value of \(\alpha\).

6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).

1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
3. the maximum speed of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_504_878_1557_632} One end of a light inextensible string of length 0.7 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg . The particle \(P\) moves in a circle on a smooth horizontal table with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string is taut and makes an angle of \(40 ^ { \circ }\) with the vertical (see diagram). Find

1. the tension in the string,
2. the force exerted on \(P\) by the table. \(P\) now moves in the same horizontal circle with constant angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\).
3. Find the maximum value of \(\omega\) for which \(P\) remains on the table.

4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.

1. Find the value of \(\theta\) at which the cone is about to topple.
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
3. Given that the solid immediately topples, find the least possible value of \(h\).