3.04a Calculate moments: about a point

\includegraphics{figure_2} A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.

1. Show that the tension in the rope attached to the pole at \(C\) is \(\left(\frac{5}{6}W + \frac{100}{3}\right)\) N. [4]
2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). [3]

Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),

1. find the value of \(W\). [3]

A steel girder \(AB\), of mass 200 kg and length 12 m, rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(AC = 2\) m and \(DB = 2\) m. A man of mass 80 kg stands on the girder at the point \(P\), where \(AP = 4\) m, as shown in Figure 1. The man is modelled as a particle and the girder is modelled as a uniform rod.

1. Find the magnitude of the reaction on the girder at the support at \(C\). [3]

The support at \(D\) is now moved to the point \(X\) on the girder, where \(XB = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find

1. the magnitude of the reaction at the support at \(X\), [2]
2. the value of \(x\). [4]

\includegraphics{figure_1} A uniform rod \(AB\) has length \(100 \text{ cm}\). Two light pans are suspended, one from each end of the rod, by two strings which are assumed to be light and inextensible. The system forms a balance with the rod resting horizontally on a smooth pivot, as shown in Fig. 1. A particle of weight \(16 \text{ N}\) is placed in the pan at \(A\) and a particle of weight \(5 \text{ N}\) is placed in the pan at \(B\). The rod rests horizontally in equilibrium when the pivot is at the point \(C\) on the rod, where \(AC = 30 \text{ cm}\).

1. Find the weight of the rod. [3]

The particle in the pan at \(A\) is replaced by a particle of weight \(3.5 \text{ N}\). The particle of weight \(5 \text{ N}\) remains in the pan at \(B\). The rod now rests horizontally in equilibrium when the pivot is moved to the point \(D\).

1. Find the distance \(AD\). [4]
2. Explain briefly where the assumption that the strings are light has been used in your answer to part (a). [1]

\includegraphics{figure_2} A plank \(AE\), of length \(6\) m and mass \(10\) kg, rests in a horizontal position on supports at \(B\) and \(D\), where \(AB = 1\) m and \(DE = 2\) m. A child of mass \(20\) kg stands at \(C\), the mid-point of \(BD\), as shown in Fig. 2. The child is modelled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate

1. the magnitude of the force exerted by the support on the plank at \(B\), [4]
2. the magnitude of the force exerted by the support on the plank at \(D\). [3]

The child now stands at a point \(F\) on the plank. The plank is in equilibrium and on the point of tilting about \(D\).

1. Calculate the distance \(DF\). [4]

\includegraphics{figure_3} A uniform beam \(AB\) has mass 12 kg and length 3 m. The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(BC = 1\) m, as shown in Figure 3. The beam is modelled as a uniform rod.

1. Find the reaction on the beam at \(C\). [3]

A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.

1. Find the distance \(AD\). [7]

\includegraphics{figure_2} A beam \(AB\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(AP = 0.3\) m and \(BQ = 0.3\) m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(PX = x\) m, \(0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

1. Show that the tension in the rope attached to the beam at \(P\) is \((588 - 350x)\) N. [3]
2. Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\). [3]
3. Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]

Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),

1. find the value of \(x\). [3]

A plank \(PQR\), of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at \(P\) and \(Q\), where \(PQ = 6\) m. A child of mass 40 kg stands on the plank at a distance of 2 m from \(P\) and a block of mass \(M\) kg is placed on the plank at the end \(R\). The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at \(P\) is equal to the force exerted on the plank by the support at \(Q\). By modelling the plank as a uniform rod, and the child and the block as particles,

1. 1. find the magnitude of the force exerted on the plank by the support at \(P\),
   2. find the value of \(M\). [10]
2. State how, in your calculations, you have used the fact that the child and the block can be modelled as particles. [1]

\includegraphics{figure_5} A uniform rod \(AB\) has length 2 m and mass 50 kg. The rod is in equilibrium in a horizontal position, resting on two smooth supports at \(C\) and \(D\), where \(AC = 0.2\) metres and \(DB = x\) metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at \(D\) is twice the magnitude of the reaction on the rod at \(C\),

1. find the value of \(x\). [6]

The support at \(D\) is now moved to the point \(E\) on the rod, where \(EB = 0.4\) metres. A particle of mass \(m\) kg is placed on the rod at \(B\), and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at \(E\) is four times the magnitude of the reaction on the rod at \(C\),

1. find the value of \(m\). [7]

A beam \(AB\) has length 15 m. The beam rests horizontally in equilibrium on two smooth supports at the points \(P\) and \(Q\), where \(AP = 2\) m and \(QB = 3\) m. When a child of mass 50 kg stands on the beam at \(A\), the beam remains in equilibrium and is on the point of tilting about \(P\). When the same child of mass 50 kg stands on the beam at \(B\), the beam remains in equilibrium and is on the point of tilting about \(Q\). The child is modelled as a particle and the beam is modelled as a non-uniform rod.

1. 1. Find the mass of the beam.
   2. Find the distance of the centre of mass of the beam from \(A\). [8]

When the child stands at the point \(X\) on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,

1. find \(AX\). [6]

\includegraphics{figure_2} A plank \(AB\) has length \(4\) m. It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass \(40\) kg and Jill has mass \(25\) kg. They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(BC = 1.6\) m, the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,

1. find the mass of the plank. [3]

They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).

1. Find the distance \(BC\) in this position. [5]
2. State how you have used the modelling assumptions that
   1. the plank is uniform,
   2. the plank is a rod,
   3. Jack and Jill are particles.
   [3]

\includegraphics{figure_2} A non-uniform rod \(AB\) has length 5 m and weight 200 N. The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1.5\) m and \(DB = 1\) m, as shown in Fig. 2. The centre of mass of \(AB\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N.

1. Show that \(50x - W = 100\). [5]

The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N.

1. Obtain another equation connecting \(W\) and \(x\). [3]
2. Calculate the value of \(x\) and the value of \(W\). [4]

\includegraphics{figure_2} A plank of wood \(AB\) has mass 10 kg and length 4 m. It rests in a horizontal position on two smooth supports. One support is at the end \(A\). The other is at the point \(C\), 0.4 m from \(B\), as shown in Figure 2. A girl of mass 30 kg stands at \(B\) with the plank in equilibrium. By modelling the plank as a uniform rod and the girl as a particle,

1. find the reaction on the plank at \(A\). [4]

The girl gets off the plank. A boulder of mass \(m\) kg is placed on the plank at \(A\) and a man of mass 80 kg stands on the plank at \(B\). The plank remains in equilibrium and is on the point of tilting about \(C\). By modelling the plank again as a uniform rod, and the man and the boulder as particles,

1. find the value of \(m\). [4]

\includegraphics{figure_2} A non-uniform plank of wood \(AB\) has length 6 m and mass 90 kg. The plank is smoothly supported at its two ends \(A\) and \(B\), with \(A\) and \(B\) at the same horizontal level. A woman of mass 60 kg stands on the plank at the point \(C\), where \(AC = 2\) m, as shown in Fig. 2. The plank is in equilibrium and the magnitudes of the reactions on the plank at \(A\) and \(B\) are equal. The plank is modelled as a non-uniform rod and the woman as a particle. Find

1. the magnitude of the reaction on the plank at \(B\), [2]
2. the distance of the centre of mass of the plank from \(A\). [5]
3. State briefly how you have used the modelling assumption that
   1. the plank is a rod,
   2. the woman is a particle.
   [2]

\includegraphics{figure_2} A ladder \(AB\), of weight \(W\) and length \(4a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\mu\). The other end \(B\) rests against a smooth vertical wall. The ladder makes an angle \(\theta\) with the horizontal, where \(\tan \theta = 2\). A load of weight \(4W\) is placed at the point \(C\) on the ladder, where \(AC = 3a\), as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium,

1. show that \(\mu = 0.35\). [6]

A second load of weight \(kW\) is now placed on the ladder at \(A\). The load of weight \(4W\) is removed from \(C\) and placed on the ladder at \(B\). The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,

1. Find the range of possible values of \(k\). [7]

\includegraphics{figure_2} A horizontal uniform rod \(AB\) has mass \(m\) and length \(4a\). The end \(A\) rests against a rough vertical wall. A particle of mass \(2m\) is attached to the rod at the point \(C\), where \(AC = 3a\). One end of a light inextensible string \(BD\) is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\), where \(D\) is vertically above \(A\). The rod is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{3}{4}\), as shown in Figure 2.

1. Find the tension in the string. [5]
2. Show that the horizontal component of the force exerted by the wall on the rod has magnitude \(\frac{5}{8}mg\). [3]

The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,

1. find the value of \(\mu\). [4]

\includegraphics{figure_2} A ladder \(AB\), of mass \(m\) and length \(4a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3m\) is fixed on the ladder at the point \(C\), where \(AC = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground. [10]

Figure 2 \includegraphics{figure_2} A wooden plank \(AB\) has mass \(4m\) and length \(4a\). The end \(A\) of the plank lies on rough horizontal ground. A small stone of mass \(m\) is attached to the plank at \(B\). The plank is resting on a small smooth horizontal peg \(C\), where \(BC = a\), as shown in Figure 2. The plank is in equilibrium making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between the plank and the ground is \(\mu\). The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle. Show that

1. the reaction of the peg on the plank has magnitude \(\frac{16}{5}mg\), [3]

1. \(\mu \geq \frac{48}{61}\). [6]

1. State how you have used the information that the peg is smooth. [1]

\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\) of mass \(m\) and length \(4a\). The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at \(C\), where \(AC = 3a\). The other end of the string is attached to the wall at \(D\), where \(AD = 2a\) and \(D\) is vertically above \(A\). The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is \(T\).

1. Show that \(T = mg\sqrt{13}\). (5)

The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at \(B\). The string breaks if the tension exceeds \(2mg\sqrt{13}\). Given that the string does not break,

1. show that \(M \leq \frac{5}{2}m\). (3)

\includegraphics{figure_3} A uniform rod \(AB\), of mass \(3m\) and length \(4a\), is held in a horizontal position with the end \(A\) against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\) vertically above \(A\), where \(AD = 3a\). A particle of mass \(3m\) is attached to the rod at \(C\), where \(AC = x\). The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is \(\frac{25}{4}mg\). Show that

1. \(x = 3a\), [5]
2. the horizontal component of the force exerted by the wall on the rod has magnitude \(5mg\). [3]

The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is about to slip,

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

A rough circular cylinder of radius \(4a\) is fixed to a rough horizontal plane with its axis horizontal. A uniform rod \(AB\), of weight \(W\) and length \(6a\sqrt{3}\), rests with its lower end \(A\) on the plane and a point \(C\) of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at 60° to the horizontal, as shown in Figure 1. \includegraphics{figure_1}

1. Show that \(AC = 4a\sqrt{3}\) [2]

The coefficient of friction between the rod and the cylinder is \(\frac{\sqrt{3}}{3}\) and the coefficient of friction between the rod and the plane is \(\mu\). Given that friction is limiting at both \(A\) and \(C\),

1. find the value of \(\mu\). [9]

\(AB\) is a light rod. Forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\), of magnitudes \(3\) N, \(2\) N and \(6\) N respectively, act upwards at right angles to the rod in a vertical plane at points dividing \(AB\) in the ratio \(1:4:2:4\), as shown. \includegraphics{figure_4} A single force \(\mathbf{P}\) is applied downwards at the point \(C\) to keep the rod horizontal in equilibrium.

1. State the magnitude of \(\mathbf{P}\). [1 mark]
2. Show that \(AC:CB = 5:6\). [5 marks]

Two particles, of weights \(3\) N and \(k\) N, are now placed on the rod at \(A\) and \(B\) respectively, while the same upward forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\) act as before. It is found that a single downward force at the same point \(C\) as before keeps \(AB\) horizontal under gravity.

1. Find the value of \(k\). [6 marks]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm.

1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]

\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).

1. Use the fact that \(OP\) is perpendicular to the line of action of \(\mathbf{R}\) to calculate the moment of \(\mathbf{R}\) about an axis through the origin and perpendicular to the \(x\)-\(y\) plane. [3 marks]

A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}

1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]

A boy holds a 30 cm metal ruler between three fingers of one hand, pushing down with the middle finger and up with the other two, at the points marked 5 cm, 10 cm and \(x\) cm and exerting forces of magnitude 11 N, 18 N and 8 N respectively. The ruler is in equilibrium in this position. Modelling the ruler as a uniform rod, find \includegraphics{figure_1}

1. the mass of the ruler, in grams, \hfill [3 marks]
2. the value of \(x\). \hfill [3 marks]
3. State how you have used the modelling assumption that the ruler is a uniform rod. \hfill [1 mark]

\includegraphics{figure_1} Figure 1 shows a uniform plank \(AB\) of length 8 m and mass 30 kg. It is supported in a horizontal position by two pivots, one situated at \(A\) and the other 2 m from \(B\). A man whose mass is 80 kg is standing on the plank 2 m from \(A\) when his dog steps onto the plank at \(B\). Given that the plank remains in equilibrium and that the magnitude of the forces exerted by each of the pivots on the plank are equal,

1. calculate the magnitude of the force exerted on the plank by the pivot at \(A\), [5 marks]
2. find the dog's mass. [3 marks]

If the dog was heavier and the plank was on the point of tilting,

1. explain how the force exerted on the plank by each of the pivots would be changed. [2 marks]