6.04e Rigid body equilibrium: coplanar forces

\includegraphics{figure_2} A uniform wire has the shape of a semicircular arc, with diameter \(AB\) of length \(0.8 \text{ m}\). The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(AB\) inclined at \(70°\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall \(0.8 \text{ m}\) vertically above \(A\). The tension in the string is \(15 \text{ N}\) (see diagram).

1. Show that the horizontal distance of the centre of mass of the wire from the wall is \(0.463 \text{ m}\), correct to 3 significant figures. [3]
2. Calculate the weight of the wire. [2]

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4 \text{ m}\) and height \(4.4 \text{ m}\). A uniform solid cylinder has radius \(0.4 \text{ m}\) and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_2} A uniform wire has the shape of a semicircular arc, with diameter \(AB\) of length \(0.8\) m. The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(AB\) inclined at \(70°\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall \(0.8\) m vertically above \(A\). The tension in the string is \(15\) N (see diagram).

1. Show that the horizontal distance of the centre of mass of the wire from the wall is \(0.463\) m, correct to 3 significant figures. [3]
2. Calculate the weight of the wire. [2]

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4\) m and height \(4.4\) m. A uniform solid cylinder has radius \(0.4\) m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_5} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).

1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]

\includegraphics{figure_3} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).

1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]

\includegraphics{figure_3} \(ABC\) is an object made from a uniform wire consisting of two straight portions \(AB\) and \(BC\), in which \(AB = a\), \(BC = x\) and angle \(ABC = 90°\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(AB\) and the horizontal is \(\theta\) (see diagram).

1. Show that \(x^2 \tan \theta - 2ax - a^2 = 0\). [3]
2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\). [2]

A cylindrical container is open at the top. The curved surface and the circular base of the container are both made from the same thin uniform material. The container has radius \(0.2 \text{ m}\) and height \(0.9 \text{ m}\).

1. Show that the centre of mass of the container is \(0.405 \text{ m}\) from the base. [3]

The container is placed with its base on a rough inclined plane. The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.

1. Find the coefficient of friction between the container and the plane. [3]

\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCD\) with \(AB = 0.75 \text{ m}\), \(AD = 0.6 \text{ m}\) and \(BC = 0.9 \text{ m}\). Angle \(BAD =\) angle \(ABC = 90°\).

1. Show that the distance of the centre of mass of the lamina from \(AB\) is \(0.38 \text{ m}\), and find the distance of the centre of mass from \(BC\). [5]

The lamina is freely suspended at \(B\) and hangs in equilibrium.

1. Find the angle between \(BC\) and the vertical. [2]

\includegraphics{figure_1} \(ABCD\) is a uniform lamina with \(AB = 1.8\) m, \(AD = DC = 0.9\) m, and \(AD\) perpendicular to \(AB\) and \(DC\) (see diagram).

1. Find the distance of the centre of mass of the lamina from \(AB\) and the distance from \(AD\). [4]

The lamina is freely suspended at \(A\) and hangs in equilibrium.

1. Calculate the angle between \(AB\) and the vertical. [2]

\includegraphics{figure_4} A uniform rod \(AB\) has weight \(15\) N and length \(1.2\) m. The end \(A\) of the rod is in contact with a rough plane inclined at \(30°\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).

1. Show that the magnitude of the force applied at \(B\) is \(4.33\) N, correct to \(3\) significant figures. [3]
2. Find the magnitude of the frictional force exerted by the plane on the rod. [2]
3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane. [3]

\includegraphics{figure_1} A non-uniform rod \(AB\), of length 0.6 m and weight 9 N, has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30°\) to the horizontal. The tension in the string is \(T\) N (see diagram).

1. Calculate \(T\). [2]
2. Find the least possible value of the coefficient of friction at \(A\). [3]

A uniform solid cylinder has radius 0.7 m and height \(h\) 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, \(θ°\), is increased gradually until the cone is about to topple.

1. Find the value of \(θ\) at which the cone is about to topple. [2]
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). [2]

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{figure_4}

1. Given that the solid immediately topples, find the least possible value of \(h\). [5]

\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\), [3]
2. the least possible value of the coefficient of friction at \(B\). [4]

A smooth sphere of mass \(M\) and radius \(a\) rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle \(\alpha\) with the horizontal.

1. Find the magnitude of each of the contact forces acting on the sphere.
2. Find the range of values of \(\alpha\) for which this equilibrium is possible.

[8]

\includegraphics{figure_7} A uniform solid hemisphere of mass \(M\) and radius \(a\) is placed with its curved surface on rough horizontal ground. A horizontal force \(P\) is applied to the hemisphere at the centre of its flat circular face.

1. Find the minimum value of the coefficient of friction \(\mu\) between the hemisphere and the ground for the hemisphere to slide without toppling.
2. Show that if \(\mu < \frac{3}{8}\), the hemisphere will topple.
3. Find the maximum horizontal distance that the centre of mass of the hemisphere moves before toppling begins, given that \(\mu = \frac{1}{4}\) and the hemisphere starts from rest.
4. Find the angular acceleration of the hemisphere about its point of contact with the ground at the instant when toppling begins.

[16]

\includegraphics{figure_2} A uniform rod \(AB\) of mass \(3m\) and length \(4a\) rests in equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\).

1. Find the normal reaction at \(A\) and the normal reaction at \(B\). [4]
2. Find the coefficient of friction between the rod and the ground. [2]

\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.

1. Find the distance of the centre of mass above the base of the cylinder. [5]
2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]

A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\,\text{m}\) and weight \(8\,\text{N}\) has a particle of weight \(2\,\text{N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\,\text{m}\) and modulus of elasticity \(20\,\text{N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\,\text{m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).

1. Show that the tension in the string is \(20\sin\theta\,\text{N}\). [1]
2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
3. Find \(\theta\). [3]

\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_2} A uniform wire is bent to form an object which has a semicircular arc with diameter \(AB\) of length 1.2 m, with a smaller semicircular arc with diameter \(BC\) 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 \(ACB\) is 0.191 m, correct to 3 significant figures. [3]

The object is freely suspended at \(A\) and hangs in equilibrium.

1. Find the angle between \(ACB\) and the vertical. [4]

\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).

1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]

The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) 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 \(BC\).

1. Given that the prism is on the point of toppling, calculate \(a\). [3]

\includegraphics{figure_2} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).

1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone. [3]
2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\). The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]