3.04b Equilibrium: zero resultant moment and force

Two forces \(\mathbf{F}_1 = (2i + j)\) N and \(\mathbf{F}_2 = (-2j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \(\mathbf{r}_1 = (3i + j + k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \(\mathbf{r}_2 = (i - 2j)\) m. A third force \(\mathbf{F}_3\) acts on the body such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are in equilibrium.

1. Find the magnitude of \(\mathbf{F}_3\). [4]

1. Find a vector equation of the line of action of \(\mathbf{F}_3\). [8]

The force \(\mathbf{F}_3\) is replaced by a fourth force \(\mathbf{F}_4\), acting through the origin \(O\), such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_4\) are equivalent to a couple.

1. Find the magnitude of this couple. [4]

Two forces \(\mathbf{F}_1 = (i + 2j + 3k)\) N and \(\mathbf{F}_2 = (3i + j + 2k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((2i + k)\) m and the force \(\mathbf{F}_2\) acts through the point with position vector \((j + 2k)\) m.

1. If the two forces are equivalent to a single force \(\mathbf{R}\), find
   1. \(\mathbf{R}\), [2]
   2. a vector equation of the line of action of \(\mathbf{R}\), in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\). [6]

1. If the two forces are equivalent to a single force acting through the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\), find the magnitude of \(\mathbf{G}\). [5]

Two forces \(\mathbf{F}_1 = (3i + k)\) N and \(\mathbf{F}_2 = (4i + j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2i - j + 3k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3i + 2k)\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2i - k)\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(R\) of mass \(2\) kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, the velocity \(\mathbf{v} \text{ m s}^{-1}\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf{v} = (pt^2 - 3t)\mathbf{i} + (8t + q)\mathbf{j}\), where \(p\) and \(q\) are constants and \(p < 0\).

1. Given that when \(t = 0.5\) the magnitude of \(\mathbf{F}\) is \(20\), find the value of \(p\). [6]

When \(t = 0\), \(R\) is at the point with position vector \((2\mathbf{i} - 3\mathbf{j})\) m.

1. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). [4]

When \(t = 1\), \(R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2\mathbf{i} - 8\mathbf{j}\).

1. Find the value of \(q\). [3]

\includegraphics{figure_10} The diagram shows a wall-mounted light. It consists of a rod \(AB\) of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at \(A\), and a lamp of mass 0.5 kg fixed at \(B\). The system is held in equilibrium by a chain \(CD\) whose end \(C\) is attached to the midpoint of \(AB\). The end \(D\) is fixed to the wall a distance 0.4 m vertically above \(A\). The rod \(AB\) makes an angle of \(60°\) with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.

1. By taking moments about \(A\), determine the tension in the chain. [4]
2. 1. Determine the magnitude of the force exerted on the rod at \(A\). [4]
   2. Calculate the direction of the force exerted on the rod at \(A\). [2]
3. Suggest one improvement that could be made to the model to make it more realistic. [1]

A beam, \(AB\), has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at \(C\), where \(AC = 0.5\) m. The other rope is attached to the beam at \(D\), where \(DB = 0.7\) m (see diagram). The beam is modelled as a non-uniform rod and the ropes as light inextensible strings. It is given that the tension in the rope at \(C\) is three times the tension in the rope at \(D\).

1. Determine the distance of the centre of mass of the beam from \(A\). [5]

A particle of mass \(m\) kg is now placed on the beam at a point where the magnitude of the moment of the particle's weight about \(C\) is 3.5\(mg\) N m. The beam remains horizontal and in equilibrium.

1. Determine the largest possible value of \(m\). [2]

\includegraphics{figure_11} A uniform rod \(AB\) of mass 4 kg and length 3 m rests in a vertical plane with \(A\) on rough horizontal ground. A particle of mass 1 kg is attached to the rod at \(B\). The rod makes an angle of \(60°\) with the horizontal and is held in limiting equilibrium by a light inextensible string \(CD\). \(D\) is a fixed point vertically above \(A\) and \(CD\) makes an angle of \(60°\) with the vertical. The distance \(AC\) is \(x\) m (see diagram).

1. Find, in terms of \(g\) and \(x\), the tension in the string. [3]

The coefficient of friction between the rod and the ground is \(\frac{9\sqrt{3}}{35}\).

1. Determine the value of \(x\). [4]

\includegraphics{figure_11} A uniform rod \(AB\), of weight \(20 \text{N}\) and length \(2.8 \text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55°\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the magnitude of the force acting on the rod at \(B\) is \(9.56 \text{N}\), correct to 3 significant figures. [3]
2. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]

A uniform rod, AB, has length 4 metres. The rod is resting on a support at its midpoint C. A particle of mass 4 kg is placed 0.6 metres to the left of C. Another particle of mass 1.5 kg is placed \(x\) metres to the right of C, as shown. \includegraphics{figure_3} The rod is balanced in equilibrium at C. Find \(x\). Circle your answer. [1 mark] 1.8 m 1.5 m 1.75 m 1.6 m

A metal rod, of mass \(m\) kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, \(B\), extends 6 cm beyond the edge of the shelf, \(A\), as shown in the diagram below. \includegraphics{figure_14}

1. The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of \(AB\). Show that \(m = 0.21\) [3 marks]
2. The object of mass 0.28 kilograms is removed. A number, \(n\), of identical objects, each of mass 0.048 kg, are hung from the rod all at a distance of 1 cm from \(B\). Find the maximum value of \(n\) such that the rod remains horizontal. [4 marks]
3. State one assumption you have made about the rod. [1 mark]

A uniform rod, \(AB\), has length \(7\) metres and mass \(4\) kilograms. The rod rests on a single fixed pivot point, \(C\), where \(AC = 2\) metres. A particle of weight \(W\) newtons is fixed at \(A\), as shown in the diagram. \includegraphics{figure_13} The system is in equilibrium with the rod resting horizontally.

1. Find \(W\), giving your answer in terms of \(g\). [2 marks]
2. Explain how you have used the fact that the rod is uniform in part (a). [1 mark]

A uniform rod is resting on two fixed supports at points \(A\) and \(B\). \(A\) lies at a distance \(x\) metres from one end of the rod. \(B\) lies at a distance \((x + 0.1)\) metres from the other end of the rod. The rod has length \(2L\) metres and mass \(m\) kilograms. The rod lies horizontally in equilibrium as shown in the diagram below. \includegraphics{figure_17} The reaction force of the support on the rod at \(B\) is twice the reaction force of the support on the rod at \(A\). Show that $$L - x = k$$ where \(k\) is a constant to be found. [4 marks]

A uniform rod, \(AB\), has length 3 metres and mass 24 kg. A particle of mass \(M\) kg is attached to the rod at \(A\). The rod is balanced in equilibrium on a support at \(C\), which is 0.8 metres from \(A\). \includegraphics{figure_11} Find the value of \(M\). [2 marks]

\includegraphics{figure_13} A step-ladder has two sides AB and AC, each of length \(4a\). Side AB has weight \(W\) and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC, are connected by a light taut rope DE, where D is on AB, E is on AC and AD = AE = \(a\). A man of weight \(4W\) stands at a point F on AB, where BF = \(x\). The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle \(\theta\) to the vertical, as shown in Fig. 13.

1. By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W\left(1 + \frac{2x}{a}\right)\tan\theta.$$ [7]

The rope is elastic with natural length \(\frac{1}{2}a\) and modulus of elasticity \(W\).

1. Show that the condition for equilibrium is that $$x = \frac{1}{2}a(8\cos\theta - \cot\theta - 1).$$ [5]

In this question you must show detailed reasoning.

1. Hence determine, in terms of \(a\), the maximum value of \(x\) for which equilibrium is possible. [5]

END OF QUESTION PAPER

\includegraphics{figure_12} The diagram shows a uniform square lamina ABCD, of weight \(W\) and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O. The vertex A rests on a smooth plane inclined at an angle of 30° to the horizontal. The vertex B rests on a smooth plane inclined at an angle of 60° to the horizontal. OA is a line of greatest slope of the plane inclined at 30° to the horizontal and OB is a line of greatest slope of the plane inclined at 60° to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac{1}{8}aW\).

1. By resolving horizontally and vertically, determine, in terms of \(W\), the magnitude of the normal contact force between the plane and the lamina at B. [6]
2. By taking moments about A, show that \(\theta\) satisfies the equation $$2(\sqrt{3} + 2)\sin\theta - 2\cos\theta = 1.$$ [5]
3. Verify that \(\theta = 22.4°\), correct to 1 decimal place. [2]

\includegraphics{figure_4} A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G N m are also applied to the lamina so that it is in equilibrium.

1. Determine the magnitude and direction of F. [4]
2. Determine the value of G. [2]

\includegraphics{figure_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).

1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
2. Determine the range in which the value of \(\theta\) must lie. [4]

The vertices of a triangular lamina, which is in the \(x\)–\(y\) plane, are at the origin O and the points A\((2, 3)\) and B\((-2, 1)\). Forces \(2\mathbf{i} + \mathbf{j}\) and \(-3\mathbf{i} + 2\mathbf{j}\) are applied to the lamina at A and B, respectively, and a force \(\mathbf{F}\), whose line of action is in the \(x\)–\(y\) plane, is applied at O. The three forces form a couple.

1. Determine the magnitude and the direction of \(\mathbf{F}\). [4]
2. Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium. [3]

\includegraphics{figure_9} Fig. 9 shows a uniform rod AB of length \(2a\) and weight \(8W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.

1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. [4] It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4a\).
2. Determine the coefficient of friction between the bar and the ring. [6]

A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of 20° with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}

1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
2. 1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
   2. Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping. [4]

The diagram shows a uniform plank \(AB\) of length 4 m supported in horizontal equilibrium by means of a central pivot. On the plank there are three objects of masses 8 kg, 2 kg and 15 kg placed in positions \(C\), \(D\) and \(E\) respectively. The distance \(AC\) is \(0 \cdot 6\) m and the distance \(AE\) is \(2 \cdot 8\) m. \includegraphics{figure_3} Find the distance \(AD\). [4]

The diagram below shows a spotlight system that consists of a symmetrical track \(XY\) that is suspended horizontally from the ceiling by means of two vertical wires. \includegraphics{figure_9} Each of the three spotlights \(A\), \(B\), \(C\) may be moved horizontally along its corresponding shaded section of the track. The system remains in equilibrium. The track may be modelled as a light uniform rod of length \(1.8\) m and the wires are fixed at a distance of \(0.4\) m from each end. Each of the spotlights may be modelled as a particle of mass \(m\) kg, positioned at the points where they are in contact with the track. The distances of the spotlights relative to the wires are given in the diagram and are such that $$0 \leqslant d_A \leqslant 0.3, \quad 0.1 \leqslant d_B \leqslant 0.9, \quad 0 \leqslant d_C \leqslant 0.3.$$

1. Given that \(T_1\) and \(T_2\) represent the tension in wires 1 and 2 respectively, show that $$T_1 = mg(2 + d_A - d_B - d_C),$$ and find a similar expression for \(T_2\). [6]
2. 1. Find the maximum possible value of \(T_1\).
   2. Without carrying out any further calculations, write down the maximum possible value of \(T_2\). Give a reason for your answer. [3]

\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).

1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]

A uniform ladder \(AB\), of weight \(150\text{N}\) and length \(4\text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight \(750\text{N}\) is standing on the ladder at a distance \(x\text{m}\) from \(A\).

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{75}{2}(2 + 5x)\text{N}\). [4]

The coefficient of friction between the ladder and the ground is \(\frac{1}{4}\).

1. Find the greatest value of \(x\) for which equilibrium is possible. [3]

A uniform ladder \(AB\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall. The ladder is inclined at an angle of \(45°\) to the horizontal. A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\). The coefficient of friction between the ladder and the wall is \(\frac{1}{4}\) and the coefficient of friction between the ladder and the ground is \(\frac{1}{2}\). The system is in limiting equilibrium. Find \(x\). [8]