3.04b Equilibrium: zero resultant moment and force

3. \begin{figure}[h] \end{figure} A wooden beam \(A B\), of mass 150 kg and length 9 m , rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(B D = 3.5 \mathrm {~m}\). A gymnast of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,

1. find the tension in the rope attached to the beam at \(C\). The gymnast at \(P\) remains on the beam at \(P\) and another gymnast, who is also modelled as a particle, stands on the beam at \(B\). The beam remains horizontal and in equilibrium. The mass of the gymnast at \(B\) is the largest possible for which the beam remains horizontal and in equilibrium.
2. Find the tension in the rope attached to the beam at \(D\).

4. A uniform plank \(A B\) has weight 80 N and length \(x\) metres. The plank rests in equilibrium horizontally on two smooth supports at \(A\) and \(C\), where \(A C = 2 \mathrm {~m}\), as shown in Fig. 2. A rock of weight 20 N is placed at \(B\) and the plank remains in equilibrium. The reaction on the plank at \(C\) has magnitude 90 N . The plank is modelled as a rod and the rock as a particle.

1. Find the value of \(x\).
2. State how you have used the model of the rock as a particle. The support at \(A\) is now moved to a point \(D\) on the plank and the plank remains in equilibrium with the rock at \(B\). The reaction on the plank at \(C\) is now three times the reaction at \(D\).
3. Find the distance \(A D\).

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 \(k\).

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is in contact with rough horizontal ground and the end \(B\) is in contact with a smooth vertical wall. The rod rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. The coefficient of friction between the rod and the ground is \(\mu\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the rod by the wall.
2. Show that \(\mu \geqslant \frac { \sqrt { 3 } } { 6 }\). A particle of mass \(k m\) is now attached to the rod at \(B\). Given that \(\mu = \frac { \sqrt { 3 } } { 5 }\) and that the rod is now in limiting equilibrium,
3. find the value of \(k\).

5. \begin{figure}[h] \end{figure} A uniform rod, of weight \(W\) and length \(16 b\), has one end freely hinged to a fixed point \(A\). The rod rests against a smooth circular cylinder, of radius \(5 b\), fixed with its axis horizontal and at the same horizontal level as \(A\). The distance of \(A\) from the axis of the cylinder is 13b, as shown in Figure 2. The rod rests in a vertical plane which is perpendicular to the axis of the cylinder.

1. Find, in terms of \(W\), the magnitude of the reaction on the rod at its point of contact with the cylinder.
2. Show that the resultant force acting on the rod at \(A\) is inclined to the vertical at an angle \(\alpha\) where \(\tan \alpha = \frac { 40 } { 73 }\)
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6. \begin{figure}[h] \end{figure} A plank \(A B\) rests in equilibrium against a fixed horizontal pole. The plank has length 4 m and weight 20 N and rests on the pole at \(C\), where \(A C = 2.5 \mathrm {~m}\). The end \(A\) of the plank rests on rough horizontal ground and \(A B\) makes an angle \(\theta\) with the ground, as shown in
Figure 2. The coefficient of friction between the plank and the ground is \(\frac { 1 } { 4 }\).
The plank is modelled as a uniform rod and the pole as a rough horizontal peg that is perpendicular to the vertical plane containing \(A B\). Given that \(\cos \theta = \frac { 4 } { 5 }\) and that the friction is limiting at both \(A\) and \(C\),

1. find the magnitude of the normal reaction on the plank at \(C\),
2. find the coefficient of friction between the plank and the pole.

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of weight \(W\) and length \(8 a\), rests in equilibrium with the end \(A\) on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is \(C\), where \(A C = 7 a\), as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle \(\alpha\) with the horizontal .

1. Show that the normal reaction of the ground on the rod at \(A\) has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is \(\mu\) and that \(\cos \alpha = \frac { 3 } { \sqrt { 10 } }\)
2. find the range of possible values of \(\mu\).
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2. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) has sides \(A B = A C = 13 a\) and \(B C = 10 a\). The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) is applied to the lamina at \(B\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A B\) vertical. The weight of the lamina is \(W\). Find \(F\) in terms of \(W\).
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6. \begin{figure}[h] \end{figure} A ladder \(A B\) has length 6 m and mass 30 kg . The ladder rests in equilibrium at \(60 ^ { \circ }\) to the horizontal with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall, as shown in Figure 3. A man of mass 70 kg stands on the ladder at the point \(C\), where \(A C = 2 \mathrm {~m}\), and the ladder remains in equilibrium. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall. The man is modelled as a particle.

1. Find the magnitude of the force exerted on the ladder by the ground. The man climbs further up the ladder. When he is at the point \(D\) on the ladder, the ladder is about to slip. Given that the coefficient of friction between the ladder and the ground is 0.4
2. find the distance \(A D\).
3. State how you have used the modelling assumption that the ladder is a rod.

5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
3. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\) (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
      1. Explain why \(A O = 13 a\)

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 \(A\).

3. A beam \(A B\) has length 15 m and mass 25 kg . The beam is smoothly supported at the point \(P\), where \(A P = 8 \mathrm {~m}\). A man of mass 100 kg stands on the beam at a distance of 2 m from \(A\) and another man stands on the beam at a distance of 1 m from \(B\). The beam is modelled as a non-uniform rod and the men are modelled as particles. The beam is in equilibrium in a horizontal position with the reaction on the beam at \(P\) having magnitude 2009 N. Find the distance of the centre of mass of the beam from \(A\).

4. \begin{figure}[h] \end{figure} A plank \(A B\) of mass 20 kg and length 8 m is resting in a horizontal position on two supports at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A package of mass 8 kg is placed on the plank at \(C\), as shown in Figure 2. The plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the package is modelled as a particle.

1. Find the magnitude of the normal reaction
   1. between the plank and the support at \(C\),
   2. between the plank and the support at \(D\).
      (6) The package is now moved along the plank to the point \(E\). When the package is at \(E\), the magnitude of the normal reaction between the plank and the support at \(C\) is \(R\) newtons and the magnitude of the normal reaction between the plank and the support at \(D\) is \(2 R\) newtons.
2. Find the distance \(A E\).
3. State how you have used the fact that the package is modelled as a particle.

7. A non-uniform rod \(A B\) has length 6 m and mass 8 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(C\) and at \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Figure 3. The magnitude of the reaction between the rod and the support at \(D\) is twice the magnitude of the reaction between the rod and the support at \(C\). The centre of mass of the rod is at \(G\), where \(A G = x \mathrm {~m}\).

1. Show that \(x = \frac { 11 } { 3 }\). The support at \(C\) is moved to the point \(F\) on the rod, where \(A F = 2 \mathrm {~m}\). A particle of mass 3 kg is placed on the rod at \(A\). The rod remains horizontal and in equilibrium. The magnitude of the reaction between the rod and the support at \(D\) is \(k\) times the magnitude of the reaction between the rod and the support at \(F\).
2. Find the value of \(k\).
   7. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-20_223_1262_127_338}
   \end{figure}

4. \begin{figure}[h] \end{figure} A boy sees a box on the end \(Q\) of a plank \(P Q\) which overhangs a swimming pool. The plank has mass 30 kg , is 5 m long and rests in a horizontal position on two bricks. The bricks are modelled as smooth supports, one acting on the rod at \(P\) and one acting on the rod at \(R\), where \(P R = 3 \mathrm {~m}\). The support at \(R\) is on the edge of the swimming pool, as shown in Figure 2. The boy has mass 40 kg and the box has mass 2.5 kg . The plank is modelled as a uniform rod and the boy and the box are modelled as particles. The boy steps on to the plank at \(P\) and begins to walk slowly along the plank towards the box.

1. Find the distance he can walk along the plank from \(P\) before the plank starts to tilt.
2. State how you have used, in your working, the fact that the box is modelled as a particle. A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(P\). The boy is then able to walk slowly along the plank to the box at the end \(Q\) without the plank tilting. The rock is modelled as a particle.
3. Find the smallest possible value of \(M\).

2. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 6 m and weight \(W\) newtons. The beam is supported in equilibrium in a horizontal position by two vertical ropes, one attached to the beam at \(A\) and the other attached to the beam at \(C\), where \(C B = 1.5 \mathrm {~m}\), as shown in Figure 1 . The centre of mass of the beam is 2.625 m from \(A\). The ropes are modelled as light strings. The beam is modelled as a non-uniform rod. Given that the tension in the rope attached at \(C\) is 20 N greater than the tension in the rope attached at \(A\),

1. find the value of \(W\).
2. State how you have used the fact that the beam is modelled as a rod.
   

4. \begin{figure}[h] \end{figure} \begin{verbatim} A metal girder \(A B\) has weight \(W\) newtons and length 6 m . The girder rests in a horizontal position on two supports \(C\) and \(D\) where \(A C = D B = 1 \mathrm {~m}\), as shown in Figure 2. When a force of magnitude 900 N is applied vertically upwards to the girder at \(A\), the girder is about to tilt about \(D\). When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). A metal girder AB has weight When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). \end{verbatim}

3. \begin{figure}[h] \end{figure} A beam \(A D C B\) has length 5 m . The beam lies on a horizontal step with the end \(A\) on the step and the end \(B\) projecting over the edge of the step. The edge of the step is at the point \(D\) where \(D B = 1.3 \mathrm {~m}\), as shown in Figure 2. When a small boy of mass 30 kg stands on the beam at \(C\), where \(C B = 0.5 \mathrm {~m}\), the beam is on the point of tilting. The boy is modelled as a particle and the beam is modelled as a uniform rod.

1. Find the mass of the beam. A block of mass \(X \mathrm {~kg}\) is now placed on the beam at \(A\).
   The block is modelled as a particle.
2. Find the smallest value of \(X\) that will enable the boy to stand on the beam at \(B\) without the beam tilting.
3. State how you have used the modelling assumption that the block is a particle in your calculations.

4. \begin{figure}[h] \end{figure} A branch \(A B\), of length 1.5 m , rests horizontally in equilibrium on two supports.
The two supports are at the points \(C\) and \(D\), where \(A C = 0.24 \mathrm {~m}\) and \(D B = 0.36 \mathrm {~m}\), as shown in Figure 1. When a force of 150 N is applied vertically upwards at \(B\), the branch is on the point of tilting about \(C\). When a force of 225 N is applied vertically downwards at \(B\), the branch is on the point of tilting about \(D\). The branch is modelled as a non-uniform rod \(A B\) of weight \(W\) newtons.
The distance from the point \(C\) to the centre of mass of the rod is \(x\) metres.
Use the model to find

1. the value of \(W\)
2. the value of \(x\)

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 30 kg and length 3 m .
The beam rests on supports at \(C\) and \(D\) where \(A C = 0.4 \mathrm {~m}\) and \(D B = 0.4 \mathrm {~m}\), as shown in Figure 4. A person of mass 55 kg stands on the beam between \(C\) and \(D\).
The person is modelled as a particle at the point \(P\), where \(C P = x\) metres and \(0 < x < 2.2\) The beam is modelled as a uniform rod resting in equilibrium in a horizontal position.
Using the model,

1. show that the magnitude of the reaction at \(C\) is \(( 686 - 245 x ) \mathrm { N }\). The magnitude of the reaction at \(C\) is four times the magnitude of the reaction at \(D\).
   Using the model,
2. find the value of \(x\) The person steps off the beam and places a package of mass \(M \mathrm {~kg}\) at \(A\).
   The package is modelled as a particle at the point \(A\).
   The beam is now on the point of tilting about \(C\).
   Using the model,
3. find the value of \(M\)

4. \begin{figure}[h] \end{figure} A plank \(A B\), of length 6 m and mass 4 kg , rests in equilibrium horizontally on two supports at \(C\) and \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\). A brick of mass 2 kg rests on the plank at \(A\) and a brick of mass 3 kg rests on the plank at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod and all bricks are modelled as particles.

1. Find the magnitude of the reaction exerted on the plank
   1. by the support at \(C\),
   2. by the support at \(D\). The 3 kg brick is now removed and replaced with a brick of mass \(x \mathrm {~kg}\) at \(B\). The plank remains horizontal and in equilibrium but the reactions on the plank at \(C\) and at \(D\) now have equal magnitude.
2. Find the value of \(x\).

6. \begin{figure}[h] \end{figure} A plank \(A B\) has length 4 m and mass 6 kg . The plank rests in a horizontal position on two supports, one at \(B\) and one at \(C\), where \(A C = 1.5 \mathrm {~m}\). A load of mass 15 kg is placed on the plank at the point \(X\), as shown in Figure 2, and the plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the load is modelled as a particle. The magnitude of the reaction on the plank at \(C\) is twice the magnitude of the reaction on the plank at \(B\).

1. Find the magnitude of the reaction on the plank at \(C\).
2. Find the distance \(A X\). The load is now moved along the plank to a point \(Y\), between \(A\) and \(C\). Given that the plank is on the point of tipping about \(C\),
3. find the distance \(A Y\).

2. \begin{figure}[h] \end{figure} A wooden beam \(A B\) has weight 140 N and length \(2 a\) metres. The beam rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = 2 \mathrm {~m}\) and \(A D = 6 \mathrm {~m}\). A block of weight 30 N is placed on the beam at \(B\) and the beam remains horizontal and in equilibrium, as shown in Figure 2. The reaction on the beam at \(D\) has magnitude 120 N . The block is modelled as a particle and the beam is modelled as a uniform rod.

1. Find the value of \(a\). The support at \(D\) is now moved to a point \(E\) on the beam and the beam remains horizontal and in equilibrium with the block at \(B\). The magnitude of the reaction on the beam at \(C\) is now equal to the magnitude of the reaction on the beam at \(E\).
2. Find the distance \(A E\).

2. \begin{figure}[h] \end{figure} A uniform wooden beam \(A B\), of mass 20 kg and length 4 m , rests in equilibrium in a horizontal position on two supports. One support is at \(C\), where \(A C = 1.6 \mathrm {~m}\), and the other support is at \(D\), where \(D B = 0.4 \mathrm {~m}\). A boy of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 1. The beam remains in equilibrium in a horizontal position. By modelling the boy as a particle and the beam as a uniform rod,

1. 1. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(C\),
   2. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(D\). The boy now starts to walk slowly along the beam towards the end \(A\).
2. Find the greatest distance he can walk from \(P\) without the beam tilting.

7. \begin{figure}[h] \end{figure} A washing line \(A B C D\) is fixed at the points \(A\) and \(D\). There are two heavy items of clothing hanging on the washing line, one fixed at \(B\) and the other fixed at \(C\). The washing line is modelled as a light inextensible string, the item at \(B\) is modelled as a particle of mass 3 kg and the item at \(C\) is modelled as a particle of mass \(M \mathrm {~kg}\). The section \(A B\) makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), the section \(B C\) is horizontal and the section \(C D\) makes an angle \(\beta\) with the horizontal, where \(\tan \beta = \frac { 12 } { 5 }\), as shown in Figure 2. The system is in equilibrium.

1. Find the tension in \(A B\).
2. Find the tension in BC.
3. Find the value of \(M\).
   

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