3.04a Calculate moments: about a point

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}

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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4. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 8 m and mass \(M \mathrm {~kg}\). The centre of mass of the beam is \(d\) metres from \(A\). The beam is supported in equilibrium in a horizontal position by two vertical light ropes. One rope is attached to the beam at \(C\), where \(A C = 2.5 \mathrm {~m}\) and the other rope is attached to the beam at \(D\), where \(D B = 2 \mathrm {~m}\), as shown in Figure 2. A gymnast, of mass 64 kg , stands on the beam at the point \(X\), where \(A X = 1.875 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(C\). The gymnast then dismounts from the beam. A second gymnast, of mass 48 kg , now stands on the beam at the point \(Y\), where \(Y B = 0.5 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(D\). The beam is modelled as a non-uniform rod and the gymnasts are modelled as particles. Find the value of \(M\).

7. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\), of mass 60 kg and length \(8 a\) metres, rests in equilibrium in a horizontal position on two vertical supports. One support is at \(C\), where \(A C = a\) metres and the other support is at \(D\), where \(D B = 2 a\) metres, as shown in Figure 2. The magnitude of the normal reaction between the beam and the support at \(D\) is three times the magnitude of the normal reaction between the beam and the support at \(C\). By modelling the beam as a non-uniform rod whose centre of mass is at a distance \(x\) metres from \(A\),

1. find an expression for \(x\) in terms of \(a\). A box of mass \(M \mathrm {~kg}\) is placed on the beam at \(E\), where \(A E = 2 a\) metres.
   The beam remains in equilibrium in a horizontal position.
   The magnitude of the normal reaction between the beam and the support at \(C\) is now equal to the magnitude of the normal reaction between the beam and the support at \(D\). By modelling the box as a particle,
2. find the value of \(M\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 5 m and mass 5 kg . The rod rests in equilibrium in a horizontal position on two supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\), as shown in Figure 2 . A particle of mass 10 kg is placed on the rod at \(A\) and a particle of mass \(M \mathrm {~kg}\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(C\).
2. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(D\).
3. Hence, or otherwise, find the range of possible values of \(M\). \includegraphics[max width=\textwidth, alt={}, center]{61cb5bce-2fad-48f0-b6a4-e9899aa0acec-14_2256_51_310_1983}

4. \begin{figure}[h] \end{figure} Figure 1 shows a beam \(A B\), of mass \(m \mathrm {~kg}\) and length 2 m , suspended by two light vertical ropes.
The ropes are attached to the points \(C\) and \(D\) on the beam, where \(A C = 0.6 \mathrm {~m}\) and \(D B = 0.2 \mathrm {~m}\) The beam is in equilibrium in a horizontal position.
A particle of mass pmkg is attached to the beam at \(A\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod.

1. Given that the tension in the rope attached at \(C\) is four times the tension in the rope attached at \(D\), use the model to find the exact value of \(p\). The particle of mass \(p m \mathrm {~kg}\) at \(A\) is removed and replaced by a particle of mass \(q m \mathrm {~kg}\) at \(A\).
   The beam remains in equilibrium in a horizontal position but is now on the point of tilting.
2. Using the model, find the exact value of \(q\)
3. State how you have used the modelling assumption that the beam is uniform.

4. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 6.5 m and mass 1.2 kg . The centre of mass of the rod is 3 m from \(A\). The rod rests on a horizontal step and overhangs the end of the step \(C\) by 1.5 m , as shown in Figure 2. The rod is perpendicular to the edge of the step.
A particle of mass 4 kg is placed on the rod at \(B\) and another particle, whose mass is \(M \mathrm {~kg}\), is placed on the rod at \(D\), where \(A D = 0.5 \mathrm {~m}\). The rod remains in equilibrium in a horizontal position.

1. Find the smallest possible value of \(M\). The particle at \(B\) and the particle at \(D\) are now removed.
   A new particle is placed on the rod at the point \(E\), where \(E B = 0.9 \mathrm {~m}\).
   The rod remains in equilibrium in a horizontal position but is on the point of tilting about \(C\).
2. Find the magnitude of the force acting on the rod at \(C\).

3. \begin{figure}[h] \end{figure} A plank \(A B\) has length 8 m and mass 12 kg . The plank rests on two supports. One support is at \(C\), where \(A C = 3 \mathrm {~m}\) and the other support is at \(D\), where \(A D = x\) metres. A block of mass 3 kg is placed on the plank at \(B\), as shown in Figure 1. The plank rests in equilibrium in a horizontal position. The magnitude of the force exerted on the plank by the support at \(D\) is twice the magnitude of the force exerted on the plank by the support at \(C\). The plank is modelled as a uniform rod and the block is modelled as a particle. Find the value of \(x\).

1. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 9 m and mass \(M \mathrm {~kg}\).
The rod rests in equilibrium in a horizontal position on two supports, one at \(C\) where \(A C = 2.5 \mathrm {~m}\) and the other at \(D\) where \(D B = 2 \mathrm {~m}\), as shown in Figure 1 . The magnitude of the force acting on the rod at \(D\) is twice the magnitude of the force acting on the \(\operatorname { rod }\) at \(C\). The centre of mass of the rod is \(d\) metres from \(A\).
Find the value of \(d\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a beam \(A B\) with weight 24 N and length 6 m .
The beam is suspended by two light vertical ropes. The ropes are attached to the points \(C\) and \(D\) on the beam where \(A C = x\) metres and \(C D = 2 \mathrm {~m}\). The tension in the rope attached to the beam at \(C\) is double the tension in the rope attached to the beam at \(D\). The beam is modelled as a uniform rod, resting horizontally in equilibrium.
Find

1. the tension in the rope attached to the beam at \(D\).
2. the value of \(x\).

4. \begin{figure}[h] \end{figure} A diving board \(A B\) consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(A C = 0.6 \mathrm {~m}\), as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

1. Find
   1. the magnitude of the force acting on the plank at \(A\),
   2. the magnitude of the force acting on the plank at \(C\). The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N .
2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\).
3. Explain how you have used the fact that the diver is modelled as a particle.
   

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1. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find

1. the magnitude of the reaction on the rod at \(Q\),
2. the distance \(A Q\).

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings.

1. Find
   1. the tension in the rope at \(C\),
   2. the tension in the rope at \(A\). A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
2. find, in terms of \(y\), an expression for the tension in the rope at \(C\). The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
3. Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.

4. \begin{figure}[h] \end{figure} A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod \(P S\) of length 2.4 m and mass 20 kg . The legs at \(Q\) and \(R\) are 0.4 m from each end of the plank, as shown in Figure 1. Two pupils, Arthur and Beatrice, sit on the plank. Arthur has mass 60 kg and sits at the middle of the plank and Beatrice has mass 40 kg and sits at the end \(P\). The plank remains horizontal and in equilibrium. By modelling the pupils as particles, find

1. the magnitude of the normal reaction between the plank and the leg at \(Q\) and the magnitude of the normal reaction between the plank and the leg at \(R\). Beatrice stays sitting at \(P\) but Arthur now moves and sits on the plank at the point \(X\). Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction between the plank and the leg at \(Q\) is now twice the magnitude of the normal reaction between the plank and the leg at \(R\),
2. find the distance \(Q X\).

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 \(\lambda\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 4 m and weight 50 N .
The rod has its end \(A\) on rough horizontal ground. The rod is held in equilibrium at an angle \(\alpha\) to the ground by a light inextensible cable attached to the rod at \(B\), as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is \(T\) newtons. Given that \(\sin \alpha = \frac { 3 } { 5 }\)

1. show that \(T = 20\) Given also that the rod is in limiting equilibrium,
2. find the value of the coefficient of friction between the rod and the ground.

5. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of mass 15 kg and length 6 m , rests with end \(A\) on rough horizontal ground. The end \(B\) of the beam rests against a rough vertical wall. The beam is inclined at \(75 ^ { \circ }\) to the ground, as shown in Figure 2.
The coefficient of friction between the beam and the wall is 0.2
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. The beam rests in limiting equilibrium.

1. Find the magnitude of the normal reaction between the beam and the wall at \(B\).
2. Find the value of \(\mu\)

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 \(A\).