3.04b Equilibrium: zero resultant moment and force

\includegraphics{figure_4} A uniform lamina of weight 15 N is in the form of a trapezium \(ABCD\) 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 \(AB\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30°\) upwards from the horizontal (see diagram). Find the tension in the string. [5]

\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_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_2} A non-uniform rod \(AB\) of length \(0.5 \text{ m}\) and weight \(8 \text{ N}\) is freely hinged to a fixed point at \(A\). The rod makes an angle of \(30°\) with the horizontal with \(B\) above the level of \(A\). The rod is held in equilibrium by a force of magnitude \(12 \text{ N}\) acting in the vertical plane containing the rod at an angle of \(30°\) to \(AB\) applied at \(B\) (see diagram). Find the distance of the centre of mass of the rod from \(A\). [3]

\includegraphics{figure_4} A uniform beam \(AB\) has length \(2 \text{ m}\) and weight \(70 \text{ N}\). The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point \(1.7 \text{ m}\) vertically above the hinge. The other end of the rope is attached to the beam at a point \(0.8 \text{ m}\) from \(A\). The rope is at right angles to \(AB\). The beam carries a load of weight \(220 \text{ N}\) at \(B\) (see diagram).

1. Find the tension in the rope. [3]
2. Find the direction of the force exerted on the beam at \(A\). [4]

\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]

\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_2} A uniform solid cone with height \(0.8\) m and semi-vertical angle \(30°\) has weight \(20\) N. The cone rests in equilibrium with a single point \(P\) of its base in contact with a rough horizontal surface, and its vertex \(V\) vertically above \(P\). Equilibrium is maintained by a force of magnitude \(F\) N acting along the axis of symmetry of the cone and applied to \(V\) (see diagram).

1. Show that the moment of the weight of the cone about \(P\) is \(6\) N m. [2]
2. Hence find \(F\). [2]

\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.

1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]

\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_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_4} A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\).

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

A uniform rod \(AC\), of weight \(W\) and length \(3l\), rests horizontally on two supports, one at \(A\) and one at \(B\), where \(AB = 2l\). A particle of weight \(2W\) is placed on the rod at a distance \(x\) from \(A\). The rod remains horizontal and in equilibrium.

1. Find the greatest possible value of \(x\). [5]

The magnitude of the reaction of the support at \(A\) is \(R\). Due to a weakness in the support at \(A\), the greatest possible value of \(R\) is \(4W\).

1. find the least possible value of \(x\). [5]

\includegraphics{figure_2} A non-uniform rod \(AB\) has length 4 m and weight 120 N. The centre of mass of the rod is at the point \(G\) where \(AG = 2.2\) m. The rod is suspended in a horizontal position by two vertical light inextensible strings, one at each end, as shown in Figure 2. A particle of weight 40 N is placed on the rod at the point \(P\), where \(AP = x\) metres. The rod remains horizontal and in equilibrium.

1. Find, in terms of \(x\),
   1. the tension in the string at \(A\), [6]
   2. the tension in the string at \(B\).
   Either string will break if the tension in it exceeds 84 N.
2. Find the range of possible values of \(x\). [4]

\includegraphics{figure_1} A diving board \(AB\) consists of a wooden plank of length 4 m and mass 30 kg. The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(AC = 0.6\) m, as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

1. Find
   1. the magnitude of the force acting on the plank at \(A\),
   2. the magnitude of the force acting on the plank at \(C\).
   [6] The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N.
2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\). [3]
3. Explain how you have used the fact that the diver is modelled as a particle. [1]

\includegraphics{figure_1} A metal girder \(AB\), of weight 1080 N and length 6 m, rests in equilibrium in a horizontal position on two supports, one at \(C\) and one at \(D\), where \(AC = 0.5\) m and \(BD = 2\) m, as shown in Figure 1. A boy of weight 400 N stands on the girder at \(B\) and the girder remains horizontal and in equilibrium. The boy is modelled as a particle and the girder is modelled as a uniform rod.

1. Find
   1. the magnitude of the reaction on the girder at \(C\),
   2. the magnitude of the reaction on the girder at \(D\).
   [6]

The boy now stands at a point \(E\) on the girder, where \(AE = x\) metres, and the girder remains horizontal and in equilibrium. Given that the magnitude of the reaction on the girder at \(D\) is now 520 N greater than the magnitude of the reaction on the girder at \(C\),

1. find the value of \(x\). [5]

\includegraphics{figure_1} A uniform rod \(AB\) has length \(2a\) and mass \(M\). The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at \(C\) and \(D\), where \(AC = \frac{2}{5}a\) and \(DB = \frac{3}{5}a\), as shown in Figure 1. A particle \(P\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the largest possible mass of the particle \(P\) [3] Given that the mass of \(P\) is \(\frac{1}{2}M\)
2. Find, in terms of \(M\) and \(g\), the tension in the string that is attached to the rod at \(C\). [3]

A beam \(AB\) has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 1\) m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\). [7]

\includegraphics{figure_1} A heavy uniform steel girder \(AB\) has length 10 m. A load of weight 150 N is attached to the girder at \(A\) and a load of weight 250 N is attached to the girder at \(B\). The loaded girder hangs in equilibrium in a horizontal position, held by two vertical steel cables attached to the girder at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 3\) m, as shown in Fig. 1. The girder is modelled as a uniform rod, the loads as particles and the cables as light inextensible strings. The tension in the cable at \(D\) is three times the tension in the cable at \(C\).

1. Draw a diagram showing all the forces acting on the girder. [2]

Find

1. the tension in the cable at \(C\), [5]
2. the weight of the girder. [2]
3. Explain how you have used the fact that the girder is uniform. [1]

\includegraphics{figure_3} A uniform rod \(AB\) has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points \(C\) and \(D\), where \(AC = 0.5\) m and \(AD = 2\) m, as shown in Fig. 3. A particle of weight \(W\) newtons is attached to the rod at a point \(E\) where \(AE = x\) metres. The rod remains in equilibrium and the magnitude of the reaction at \(C\) is now twice the magnitude of the reaction at \(D\).

1. Show that \(W = \frac{60}{1-x}\). [8]
2. Hence deduce the range of possible values of \(x\). [2]