3.04b Equilibrium: zero resultant moment and force

\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string \(BD\) is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The pole is modelled as a uniform rod.

1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]

The vertical component of the force exerted by the wall on the pole is \(\frac{7}{4}W\). Find

1. \(x\) in terms of \(a\), [3]
2. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]

\includegraphics{figure_1} A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{2}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

\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]

\includegraphics{figure_2} A uniform rod \(AB\), of mass \(20\) kg and length \(4\) m, rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is \(0.5\). Find the magnitude of the normal reaction of the ground on the rod at \(A\). [7]

[The centre of mass of a semi-circular lamina of radius \(r\) is \(\frac{4r}{3\pi}\) from the centre] \includegraphics{figure_3} A template \(T\) consists of a uniform plane lamina \(PQRQS\), as shown in Figure 3. The lamina is bounded by two semicircles, with diameters \(SQ\) and \(QR\), and by the sides \(SP\), \(PQ\) and \(QR\) of the rectangle \(PQRS\). The point \(O\) is the mid-point of \(SR\), \(PQ = 12\) cm and \(QR = 2\) cm.

1. Show that the centre of mass of \(T\) is a distance \(\frac{4|2x^2 - 3|}{8x + 3\pi}\) cm from \(SR\). [7]

The template \(T\) is freely suspended from the point \(P\) and hangs in equilibrium. Given that \(x = 2\) and that \(\theta\) is the angle that \(PQ\) makes with the horizontal,

1. show that \(\tan \theta = \frac{48 + 9\pi}{22 + 6\pi}\). [4]

\includegraphics{figure_2} A uniform rod \(AB\) has mass \(4\) kg and length \(1.4\) m. The end \(A\) is resting on rough horizontal ground. A light string \(BC\) has one end attached to \(B\) and the other end attached to a fixed point \(C\). The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at \(20°\) to the ground, as shown in Figure 2.

1. Find the tension in the string. [4]

Given that the rod is about to slip,

1. find the coefficient of friction between the rod and the ground. [7]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5. The other end \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of 30° with the wall. A man of mass \(5m\) stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(ka\). Find the value of \(k\). [9]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6. The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{5}{8}W\). [6]
2. Find the value of \(\beta\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

\includegraphics{figure_2} A uniform steel girder \(AB\), of mass 40 kg and length 3 m, is freely hinged at \(A\) to a vertical wall. The girder is supported in a horizontal position by a steel cable attached to the girder at \(B\). The other end of the cable is attached to the point \(C\) vertically above \(A\) on the wall, with \(\angle ABC = \alpha\), where \(\tan \alpha = \frac{4}{3}\). A load of mass 60 kg is suspended by another cable from the girder at the point \(D\), where \(AD = 2\) m, as shown in Fig. 2. The girder remains horizontal and in equilibrium. The girder is modelled as a rod, and the cables as light inextensible strings.

1. Show that the tension in the cable \(BC\) is 980 N. [5]
2. Find the magnitude of the reaction on the girder at \(A\). [6]
3. Explain how you have used the modelling assumption that the cable at \(D\) is light. [1]

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]

\includegraphics{figure_3} A uniform rod \(AB\) has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.

1. Show that the normal reaction at \(P\) has magnitude 25 N. [4]
2. Find the magnitude of the force on the rod at \(A\). [4]

The coefficient of friction between the rod and the peg is \(\mu\).

1. Find the range of possible values of \(\mu\). [4]

\includegraphics{figure_1} A rod \(AB\), of mass \(2m\) and length \(2a\), is suspended from a fixed point \(C\) by two light strings \(AC\) and \(BC\). The rod rests horizontally in equilibrium with \(AC\) making an angle \(\alpha\) with the rod, where \(\tan \alpha = \frac{3}{4}\), and with \(AC\) perpendicular to \(BC\), as shown in Fig. 1.

1. Give a reason why the rod cannot be uniform. [1]
2. Show that the tension in \(BC\) is \(\frac{4}{5}mg\) and find the tension in \(AC\). [5]

The string \(BC\) is elastic, with natural length \(a\) and modulus of elasticity \(kmg\), where \(k\) is constant.

1. Find the value of \(k\). [4]

\includegraphics{figure_2} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre \(O\) of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is \(r\) and the radius of the base of the cone is \(2r\). The height of the cone and the height of the cylinder are each \(h\). The centre of mass of the model is at the point \(G\).

1. Show that \(OG = \frac{1}{14}h\). [8]

The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{r}{7h}\). The desk top is rough enough to prevent slipping and the model is about to topple.

1. Find \(r\) in terms of \(h\). [4]

\includegraphics{figure_3} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac{1}{4}(x - 2)^2\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid \(S\) is made by rotating \(R\) through \(360°\) about the \(x\)-axis. Using integration,

1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\), [4]
2. show that the centre of mass of \(S\) is \(\frac{4}{5}\) cm from its plane face. [7]

\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm. One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10W\) newtons and the weight of \(S\) is \(2W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.

1. Find the magnitude of the force of the support \(A\) on the tool. [5]

In a theatre, three lights \(A\), \(B\) and \(C\) are suspended from a horizontal beam \(XY\) of length 4.5 m. \(A\) and \(C\) are each of mass 8 kg and \(B\) is of mass 6 kg. The beam \(XY\) is held in place by vertical ropes \(PX\) and \(QY\), as shown. \includegraphics{figure_8} In a simple mathematical model of this situation, \(XY\) is modelled as a light rod.

1. Calculate the tension in each of \(PX\) and \(QY\). [6 marks]

In a refined model, \(XY\) is modelled as a uniform rod of mass \(m\) kg.

1. If the tension in \(PX\) is 1.5 times that in \(QY\), calculate the value of \(m\). [6 marks]

A plank of wood \(AB\), of mass 8 kg and length 6 m, rests on a support at \(P\), where \(AP = 4\) m. When particles of mass 1 kg and \(k\) kg are suspended from \(A\) and \(B\) respectively, the plank rests horizontally in equilibrium. Modelling the plank as a uniform rod, find

1. the value of \(k\), [3 marks]
2. the magnitude of the force exerted by the support on the plank at \(P\). [2 marks]

An iron bar \(AB\), of length \(4\) m, is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(PB = 0.85\) m. The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(AG = 1.4\) m, and the wire is modelled as a light inextensible string. Given that the tension in the wire is \(12\) N, calculate

1. the weight of the bar, \hfill [4 marks]
2. the magnitude of the reaction on the bar at \(A\). \hfill [2 marks]
3. State briefly how you have used the given modelling assumption about the bar. \hfill [1 mark]

Two particles \(P\) and \(Q\), of masses \(3\) kg and \(2\) kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30°\), \(60°\) and \(90°\), as shown. \(P\) and \(Q\) are connected by a light string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge. \includegraphics{figure_6} The system is released from rest with \(P\) \(0.8\) m from the pulley and \(Q\) \(1\) m from the bottom of the wedge, and \(Q\) starts to move down. Calculate

1. the acceleration of either particle, \hfill [5 marks]
2. the tension in the string, \hfill [2 marks]
3. the speed with which \(P\) reaches the pulley. \hfill [3 marks]

Two modelling assumptions have been made about the string and the pulley.

1. State these two assumptions and briefly describe how you have used each one in your solution. \hfill [4 marks]

\(AB\) 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]

A uniform plank \(XY\) has length 7 m and mass 2 kg. It is placed with the portion \(ZY\) in contact with a horizontal surface, where \(ZY = 2.8\) m. To prevent the plank from toppling, a stone is placed on the plank at \(Y\). \includegraphics{figure_2}

1. Find the smallest possible mass of the stone. [4 marks]
2. State, with a reason, whether your answer to part (a) would be greater or smaller if a shorter portion of the plank were in contact with the surface. [2 marks]