Beam suspended by vertical ropes

5. A steel girder \(A B\) has weight 210 N . It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end \(A\). The other cable is attached to the point \(C\) on the girder, where \(A C = 90 \mathrm {~cm}\), as shown in Figure 3. The girder is modelled as a uniform rod, and the cables as light inextensible strings. Given that the tension in the cable at \(C\) is twice the tension in the cable at \(A\), find

1. the tension in the cable at \(A\),
2. show that \(A B = 120 \mathrm {~cm}\). A small load of weight \(W\) newtons is attached to the girder at \(B\). The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at \(C\) is now three times the tension in the cable at \(A\).
3. Find the value of \(W\).

4. \begin{figure}[h] \end{figure} A beam \(A B\) has weight \(W\) newtons and length 4 m . The beam is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\) and the other rope is attached to the point \(C\) on the beam, where \(A C = d\) metres, as shown in Figure 3. The beam is modelled as a uniform rod and the ropes as light inextensible strings. The tension in the rope attached at \(C\) is double the tension in the rope attached at \(A\).

1. Find the value of \(d\). A small load of weight \(k W\) newtons is attached to the beam at \(B\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. The tension in the rope attached at \(C\) is now four times the tension in the rope attached at \(A\).
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} A beam \(A B\) has length 5 m and mass 25 kg . The beam is suspended in equilibrium in a horizontal position by two vertical ropes. One rope is attached to the beam at \(A\) and the other rope is attached to the point \(C\) on the beam where \(C B = 0.5 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 60 kg is attached to the beam at \(B\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod and the ropes are modelled as light strings.

1. Find
   1. the tension in the rope attached to the beam at \(A\),
   2. the tension in the rope attached to the beam at \(C\). Particle \(P\) is removed and replaced by a particle \(Q\) of mass \(M \mathrm {~kg}\) at \(B\). Given that the beam remains in equilibrium in a horizontal position,
2. find
   1. the greatest possible value of \(M\),
   2. the greatest possible tension in the rope attached to the beam at \(C\).

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. \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.

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 A uniform plank \(A B\), of length 6 m , has mass 25 kg . It is supported in equilibrium in a horizontal position by two vertical inextensible ropes. One of the ropes is attached to the plank at the point \(P\) and the other rope is attached to the plank at the point \(Q\), where \(A P = 1 \mathrm {~m}\) and \(Q B = 0.8 \mathrm {~m}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-2_227_1187_2252_424}

1. 1. Find the tension in each rope.
   2. State how you have used the fact that the plank is uniform in your solution. (1 mark)
2. A particle of mass \(m \mathrm {~kg}\) is attached to the plank at point \(B\), and the tension in each rope is now the same. Find \(m\).

1 A uniform beam, \(A B\), has mass 20 kg and length 7 metres. A rope is attached to the beam at \(A\). A second rope is attached to the beam at the point \(C\), which is 2 metres from \(B\). Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.

2 A hotel sign consists of a uniform rectangular lamina of weight \(W\). The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points \(A\) and \(B\), as shown in the diagram. The edge containing \(A\) and \(B\) is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively.

1. Draw a diagram to show the forces acting on the sign.
2. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(W\).
3. Explain how you have used the fact that the lamina is uniform in answering part (b).

2 A uniform plank, of length 6 metres, has mass 40 kg . The plank is held in equilibrium in a horizontal position by two vertical ropes attached to the plank at \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-2_323_1162_1464_440}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the tension in the rope attached to the plank at \(B\) is \(21 g \mathrm {~N}\).
3. Find the tension in the rope that is attached to the plank at \(A\).
4. State where in your solution you have used the fact that the plank is uniform.

\includegraphics{figure_2} A non-uniform rod \(AB\) has length 4 m and weight 120 N. The centre of mass of the rod is at the point \(G\) where \(AG = 2.2\) m. The rod is suspended in a horizontal position by two vertical light inextensible strings, one at each end, as shown in Figure 2. A particle of weight 40 N is placed on the rod at the point \(P\), where \(AP = x\) metres. The rod remains horizontal and in equilibrium.

1. Find, in terms of \(x\),
   1. the tension in the string at \(A\), [6]
   2. the tension in the string at \(B\).
   Either string will break if the tension in it exceeds 84 N.
2. Find the range of possible values of \(x\). [4]

\includegraphics{figure_1} A heavy uniform steel girder \(AB\) has length 10 m. A load of weight 150 N is attached to the girder at \(A\) and a load of weight 250 N is attached to the girder at \(B\). The loaded girder hangs in equilibrium in a horizontal position, held by two vertical steel cables attached to the girder at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 3\) m, as shown in Fig. 1. The girder is modelled as a uniform rod, the loads as particles and the cables as light inextensible strings. The tension in the cable at \(D\) is three times the tension in the cable at \(C\).

1. Draw a diagram showing all the forces acting on the girder. [2]

Find

1. the tension in the cable at \(C\), [5]
2. the weight of the girder. [2]
3. Explain how you have used the fact that the girder is uniform. [1]

\includegraphics{figure_1} A plank \(AB\) has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(AB\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate

1. the tension in the rope at \(C\), [2]
2. the distance \(CB\). [5]

\includegraphics{figure_2} A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.

1. Show that the tension in the rope attached to the pole at \(C\) is \(\left(\frac{5}{6}W + \frac{100}{3}\right)\) N. [4]
2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). [3]

Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),

1. find the value of \(W\). [3]

\includegraphics{figure_2} A beam \(AB\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(AP = 0.3\) m and \(BQ = 0.3\) m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(PX = x\) m, \(0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

1. Show that the tension in the rope attached to the beam at \(P\) is \((588 - 350x)\) N. [3]
2. Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\). [3]
3. Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]

Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),

1. find the value of \(x\). [3]

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 uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}

1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]

\includegraphics{figure_1} Figure 1 shows two window cleaners, Alan and Baber, of mass 60 kg and 100 kg respectively standing on a platform \(PQ\) of length 3 metres and mass 20 kg. The platform is suspended by two vertical cables attached to the ends \(P\) and \(Q\). Alan is standing at the point \(A\), 1.25 metres from \(P\), Baber is standing at the point \(B\) and the tension in the cable at \(P\) is twice the tension in the cable at \(Q\). Modelling the platform as a uniform rod and Alan and Baber as particles,

1. find the tension in the cable at \(P\), [2 marks]
2. find the distance \(BP\). [5 marks]
3. State how you have used the modelling assumptions that
   1. the platform is uniform,
   2. the platform is a rod.
   [2 marks]

A beam, \(AB\), has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at \(C\), where \(AC = 0.5\) m. The other rope is attached to the beam at \(D\), where \(DB = 0.7\) m (see diagram). The beam is modelled as a non-uniform rod and the ropes as light inextensible strings. It is given that the tension in the rope at \(C\) is three times the tension in the rope at \(D\).

1. Determine the distance of the centre of mass of the beam from \(A\). [5]

A particle of mass \(m\) kg is now placed on the beam at a point where the magnitude of the moment of the particle's weight about \(C\) is 3.5\(mg\) N m. The beam remains horizontal and in equilibrium.

1. Determine the largest possible value of \(m\). [2]