Rod hinged to wall with string support

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(k m\) is fixed to the rod at \(B\). The rod is held in equilibrium, at an angle \(\theta\) to the horizontal, by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = \frac { 5 } { 4 } a\), as shown in Figure 2. The line of action of the force at \(C\) is at right angles to \(A B\) and in the vertical plane containing \(A B\). Given that \(\tan \theta = \frac { 3 } { 4 }\)

1. show that \(F = \frac { 16 } { 25 } m g ( 1 + 2 k )\),
2. find, in terms of \(m , g\) and \(k\),
   1. the horizontal component of the force exerted by the hinge on the rod at \(A\),
   2. the vertical component of the force exerted by the hinge on the rod at \(A\). Given also that the force acting on the rod at \(A\) acts at \(45 ^ { \circ }\) above the horizontal,
3. find the value of \(k\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a uniform rod \(A B\) 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 \(\mathrm { AC } = 3 \mathrm { a }\). The other end of the string is attached to the wall at D , where \(\mathrm { AD } = 2 \mathrm { a }\) 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 \(\mathrm { T } = \mathrm { mg } \sqrt { } 13\).
   (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 \(2 \mathrm { mg } \sqrt { } 13\). Given that the string does not break,
2. show that \(M \leqslant \frac { 5 } { 2 } m\).

5. \begin{figure}[h] \end{figure} A uniform beam \(A B\) of mass 2 kg is freely hinged at one end \(A\) to a vertical wall. The beam is held in equilibrium in a horizontal position by a rope which is attached to a point \(C\) on the beam, where \(A C = 0.14 \mathrm {~m}\). The rope is attached to the point \(D\) on the wall vertically above \(A\), where \(\angle A C D = 30 ^ { \circ }\), as shown in Figure 3. The beam is modelled as a uniform rod and the rope as a light inextensible string. The tension in the rope is 63 N . Find

1. the length of \(A B\),
2. the magnitude of the resultant reaction of the hinge on the beam at \(A\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(m\) is attached to the rod at \(B\). The rod is held in equilibrium at an angle \(\theta\) to the horizontal by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = b\), as shown in Figure 3. The force at \(C\) acts at right angles to \(A B\) and in the vertical plane containing \(A B\).

1. Show that \(F = \frac { 3 a m g \cos \theta } { b }\).
2. Find, in terms of \(a , b , g , m\) and \(\theta\),
   1. the horizontal component of the force acting on the rod at \(A\),
   2. the vertical component of the force acting on the rod at \(A\). Given that the force acting on the rod at \(A\) acts along the rod,
3. find the value of \(\frac { a } { b }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).

1. Show that the tension in the string is 4.5 N .
2. Find the magnitude and direction of the force acting on the rod at \(A\).

5. A uniform rod \(X Y\), of length \(2 a\) and mass \(m\), is connected to a vertical wall by a smooth hinge at the end \(X\). A horizontal light inelastic string connects the mid-point of \(X Y\) to the wall and the rod is in equilibrium in this position.

1. Draw a diagram to show all the forces acting on the rod. Given that the tension in the horizontal string is of magnitude \(2 m g\),
2. find the angle which \(X Y\) makes with the vertical.

2 Two heavy rods AB and BC are freely jointed together at B and to a wall at A . AB has weight 90 N and centre of mass at \(\mathrm { P } ; \mathrm { BC }\) has weight 75 N and centre of mass at Q . The lengths of the rods and the positions of P and Q are shown in Fig. 2.1, with the lengths in metres. Initially, AB and BC are horizontal. There is a support at R , as shown in Fig. 2.1. The system is held in equilibrium by a vertical force acting at C . \begin{figure}[h] \end{figure}

1. Draw diagrams showing all the forces acting on \(\operatorname { rod } \mathrm { AB }\) and on \(\operatorname { rod } \mathrm { BC }\). Calculate the force exerted on AB by the hinge at B and hence the force required at C . The rods are now set up as shown in Fig. 2.2. AB and BC are each inclined at \(60 ^ { \circ }\) to the vertical and C rests on a rough horizontal table. Fig. 2.3 shows all the forces acting on AB , including the forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) due to the hinge at A and the forces \(U \mathrm {~N}\) and \(V \mathrm {~N}\) in the hinge at B . The rods are in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_393_661_1615_429} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_355_438_1530_1178} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
2. By considering the equilibrium of \(\operatorname { rod } \mathrm { AB }\), show that \(60 \sqrt { 3 } = U + V \sqrt { 3 }\).
3. Draw a diagram showing all the forces acting on rod BC .
4. Find a further equation connecting \(U\) and \(V\) and hence find their values. Find also the frictional force at C .

2 Any non-exact answers to this question should be given correct to four significant figures.
A thin, straight beam AB is 2 m long. It has a weight of 600 N and its centre of mass G is 0.8 m from end A. The beam is freely pivoted about a horizontal axis through A. The beam is held horizontally in equilibrium.
Initially this is done by means of a support at B, as shown in Fig.2.1. \begin{figure}[h] \end{figure}

1. Calculate the force on the beam due to the support at B . The support at B is now removed and replaced by a wire attached to the beam 0.3 m from A and inclined at \(50 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam is still horizontal and in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_275_803_1226_671} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate the tension in the wire. The forces acting on the beam at A due to the hinge are a horizontal force \(X \mathrm {~N}\) in the direction AB , and a downward vertical force \(Y \mathrm {~N}\), which have a resultant of magnitude \(R\) at \(\alpha\) to the horizontal.
3. Calculate \(X , Y , R\) and \(\alpha\). The dotted lines in Fig. 2.3 are the lines of action of the tension in the wire and the weight of the beam. These lines of action intersect at P . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_460_791_2074_678} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
4. State with a reason the size of the angle GAP.

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of mass 2 kg and length \(2 a\). The end \(A\) is attached by a smooth hinge to a fixed point on a vertical wall so that the rod can rotate freely in a vertical plane. A mass of 6 kg is placed at \(B\) and the rod is held in a horizontal position by a light string joining the midpoint of the rod to a point \(C\) on the wall, vertically above \(A\). The string is inclined at an angle of \(60 ^ { \circ }\) to the wall.

1. Show that the tension in the string is \(28 g\).
2. Find the magnitude and direction of the force exerted by the hinge on the rod, giving your answers correct to 3 significant figures.

2 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-2_515_1065_861_541} Two uniform rods \(A B\) and \(B C\), each of length 2 m , are freely jointed at \(B\). The weights of the rods are \(W \mathrm {~N}\) and 50 N respectively. The end \(A\) of \(A B\) is hinged at a fixed point. The rods \(A B\) and \(B C\) make angles \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)\) and \(\beta\) respectively with the downward vertical, and are held in equilibrium in a vertical plane by a horizontal force of magnitude 75 N acting at \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), show that \(\tan \beta = 3\).
2. Write down the horizontal and vertical components of the force acting on \(A B\) at \(B\).
3. Find the value of \(W\).

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Two uniform rods \(A B\) and \(B C\), of weights 70 N and 110 N respectively, are freely jointed at \(B\). The rods are in equilibrium in a vertical plane with \(A\) and \(C\) at the same horizontal level and \(A C = 2 \mathrm {~m}\). The \(\operatorname { rod } A B\) is freely jointed to a fixed point at \(A\) and the rod \(B C\) is freely jointed to a fixed point at \(C\). The horizontal distance between \(B\) and \(A\) is 4 m and \(B\) is 4 m below \(A C\); angle \(B A C\) is obtuse (see Fig. 1). The force exerted on the \(\operatorname { rod } A B\) at \(B\), by the \(\operatorname { rod } B C\), has horizontal and vertical components as shown in Fig. 2.

1. By taking moments about \(A\) for the \(\operatorname { rod } A B\) find the value of \(X - Y\).
2. By taking moments about \(C\) for the rod \(B C\) show that \(2 X - 3 Y + 165 = 0\).
3. Find the magnitude of the force acting between \(A B\) and \(B C\) at \(B\).

4. The diagram below shows a uniform rod \(A B\), of weight 10 N , hinged to a vertical wall at \(A\). The rod is held in a horizontal position by means of a light inextensible string. One end of the string is attached to a point \(C\) on the rod and the other end is attached to a point \(D\) on the wall. The point \(D\) is 0.6 m vertically above \(A\) and the length of \(A C\) is 0.8 m . A particle of weight 25 N is attached to the rod at \(B\) and the tension in the string is 75 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-4_572_808_612_625}

1. Find the length of the rod \(A B\).
2. Calculate the magnitude and direction of the reaction at the hinge at \(A\).

\includegraphics{figure_5} Two uniform rods, \(AB\) and \(BC\), each have length \(2a\) and weight \(W\). They are smoothly jointed at \(B\), and \(A\) is attached to a smooth fixed pivot. A light inextensible string of length \((2\sqrt{2})a\) joins \(A\) to \(C\) so that angle \(ABC = 90°\). The system hangs in equilibrium, with \(AB\) making an angle \(\alpha\) with the vertical (see diagram). By taking moments about \(A\) for the system, or otherwise, show that \(\alpha = 18.4°\), correct to the nearest \(0.1°\). [3] Find the tension in the string in the form \(kW\), giving the value of \(k\) correct to 3 significant figures. [3] Find, in terms of \(W\), the magnitude of the force acting on the rod \(BC\) at \(B\). [6]

\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_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]

\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)

A non-uniform beam \(AB\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(AG = 2.5\) m. The beam is freely suspended from its end \(A\) and is held in a horizontal position by means of a wire attached to the end \(B\). The wire makes an angle of \(20°\) with the vertical and the tension is \(T\) N (see diagram).

1. Calculate \(T\). [3]
2. Calculate the magnitude and the direction of the force acting on the beam at \(A\). [7]

\includegraphics{figure_2} Two uniform rods, \(AB\) and \(BC\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is \(1.5\) m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is \(3\) m (see diagram). The weight of \(AB\) is \(80\) N and the frictional force acting on \(AB\) at \(A\) is \(14\) N.

1. Write down the horizontal component of the force acting on \(AB\) at \(B\) and show that the vertical component of this force is \(33\) N upwards. [4]
2. Given that the force acting on \(BC\) at \(C\) has magnitude \(50\) N, find the weight of \(BC\). [4]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\), each of length \(2L\), are freely jointed at \(B\), and \(AB\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of rod \(BC\) is \(75\) N, and the tension in the string is \(50\) N (see diagram).

1. Show that \(\tan \beta = \frac{1}{3}\). [3]
2. Given that \(\tan \alpha = \frac{12}{5}\), find the weight of \(AB\). [5]

\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]