Rod hinged to wall with string support

4 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_737_700_264_721} A uniform beam has length 2.4 m and weight 68 N . The beam is hinged at a fixed point of a vertical wall, and held in a horizontal position by a light rod of length 2.5 m . One end of the rod is attached to the beam at a point 0.7 m from the wall, and the other end of the rod is attached to the wall at a point vertically below the hinge. The beam carries a load of 750 N at its end (see diagram).

1. Find the force in the rod. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally towards the wall and \(Y \mathrm {~N}\) vertically downwards.
2. Find the values of \(X\) and \(Y\).

1 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_518_609_255_769} A uniform \(\operatorname { rod } A B\) of weight 16 N is freely hinged at \(A\) to a fixed point. A force of magnitude 4 N acting perpendicular to the rod is applied at \(B\) (see diagram). Given that the rod is in equilibrium,

1. calculate the angle the rod makes with the horizontal,
2. find the magnitude and direction of the force exerted on the rod at \(A\).

2 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-2_333_737_762_705} A uniform rod \(A B\) of length 1.2 m and weight 30 N is in equilibrium with the end \(A\) in contact with a vertical wall. \(A B\) is held at right angles to the wall by a light inextensible string. The string has one end attached to the rod at \(B\) and the other end attached to a point \(C\) of the wall. The point \(C\) is 0.5 m vertically above \(A\) (see diagram). Find

1. the tension in the string,
2. the horizontal and vertical components of the force exerted on the rod by the wall at \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).

1. Find the tension in the string. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally away from the wall and \(Y \mathrm {~N}\) vertically downwards.
2. Find the values of \(X\) and \(Y\).

6. Figure 3 A uniform pole \(A B\), of weight 50 N and length 6 m , has a particle of weight \(W\) newtons attached at its end \(B\). The pole has its end \(A\) freely hinged to a vertical wall.
A light rod holds the particle and pole in equilibrium with the pole at \(60 ^ { \circ }\) to the wall. One end of the light rod is attached to the pole at \(C\), where \(A C = 4 \mathrm {~m}\).
The other end of the light rod is attached to the wall at the point \(D\).
The point \(D\) is vertically below \(A\) with \(A D = 4 \mathrm {~m}\), as shown in Figure 3.
The pole and the light rod lie in a vertical plane which is perpendicular to the wall.
The pole is modelled as a rod.
Given that the thrust in the light rod is \(60 \sqrt { 3 } \mathrm {~N}\),

1. show that \(W = 15\)
2. find the magnitude of the resultant force acting on the pole at \(A\).

5. \begin{figure}[h] \end{figure} A uniform rod, of length \(8 a\) and mass \(M\), has one end freely hinged to a fixed point \(A\) on a vertical wall. One end of a light inextensible string is attached to the rod at the point \(B\), where \(A B = 5 a\). The other end of the string is attached to the wall at the point \(C\), where \(A C = 5 a\) and \(C\) is vertically above \(A\). The rod rests in equilibrium in a vertical plane perpendicular to the wall with angle \(B A C = 70 ^ { \circ }\), as shown in Figure 3.

1. Find, in terms of \(M\) and \(g\), the tension in the string. The magnitude of the force acting on the rod at \(A\) is \(\lambda M g\), where \(\lambda\) is a constant.
2. Find, to 2 significant figures, the value of \(\lambda\).

6. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to a fixed point on a vertical wall.
A particle of weight \(\frac { 1 } { 4 } W\) is attached to the rod at \(B\).
A light inelastic string of length \(5 a\) has one end attached to the rod at the point \(C\), where \(A C = 5 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is above \(A\) and \(A D = 5 a\), as shown in Figure 4. The rod rests in equilibrium.
The tension in the string is \(T\).

1. Show that \(T = \frac { 6 } { 5 } \mathrm {~W}\)
2. Find, in terms of \(W\), the magnitude of the force exerted on the rod by the hinge at \(A\).

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.

1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { \mathbf { R } }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
2. Find
   1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
   2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).

3. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 5 kg and length 4 m , has its end \(A\) smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of \(25 ^ { \circ }\) above the horizontal by a force of magnitude \(F\) newtons applied to its end \(B\). The force acts in the vertical plane containing the rod and in a direction which makes an angle of \(40 ^ { \circ }\) with the rod, as shown in Figure 1.

1. Find the value of \(F\).
2. Find the magnitude and direction of the vertical component of the force acting on the rod at \(A\).

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of weight \(W\) has its end \(A\) freely hinged to a point on a fixed vertical wall. The rod is held in equilibrium, at angle \(\theta\) to the horizontal, by a force of magnitude \(P\). The force acts perpendicular to the rod at \(B\) and in the same vertical plane as the rod, as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall. The magnitude of the vertical component of the force exerted on the rod by the wall at \(A\) is \(Y\).

1. Show that \(Y = \frac { W } { 2 } \left( 2 - \cos ^ { 2 } \theta \right)\). Given that \(\theta = 45 ^ { \circ }\)
2. find the magnitude of the force exerted on the rod by the wall at \(A\), giving your answer in terms of \(W\).

6. \begin{figure}[h] \end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find

1. the thrust in the rod \(C D\),
2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.

1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { R }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
2. Find
   1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
   2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).

6. \begin{figure}[h] \end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find

1. the thrust in the rod \(C D\),
2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.

3 \includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-2_599_677_1151_696} A uniform beam \(A B\) of mass 15 kg and length 4 m is freely hinged to a vertical wall at \(A\). The beam is held in equilibrium in a horizontal position by a light rod \(P Q\) of length \(1.5 \mathrm {~m} . P\) is fixed to the wall vertically below \(A\) and \(P Q\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram). The force exerted on the beam at \(Q\) by the rod is in the direction \(P Q\). Find

1. the magnitude of the force exerted on the beam at \(Q\),
2. the magnitude and direction of the force exerted on the beam at \(A\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length \(2 a\) and mass 8 kg is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium inclined at an angle of \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(F\) newtons acting horizontally at \(B\) which is below the level of \(A\) as shown in Figure 2.

1. Find, correct to 3 significant figures, the value of \(F\).
2. Show that the magnitude of the reaction at the hinge is 133 N , correct to 3 significant figures, and find to the nearest degree the acute angle which the reaction makes with the vertical.