Uniform beam on two supports

3. \begin{figure}[h] \end{figure} A uniform beam \(A B\) has mass 20 kg and length 6 m . The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at \(C\), where \(A C = 1 \mathrm {~m}\), and the other is at the end \(B\), as shown in Figure 1. The beam is modelled as a rod.

1. Find the magnitudes of the reactions on the beam at \(B\) and at \(C\). A boy of mass 30 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The magnitudes of the reactions on the beam at \(B\) and at \(C\) are now equal. The boy is modelled as a particle.
2. Find the distance \(A D\).

1. \begin{figure}[h] \end{figure} A uniform plank \(A B\) has mass 40 kg and length 4 m . It is supported in a horizontal position by two smooth pivots, one at the end \(A\), the other at the point \(C\) of the plank where \(A C = 3 \mathrm {~m}\), as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to \(R\) newtons. By modelling the plank as a rod and the man as a particle, find

1. the value of \(R\),
2. the distance of the man from \(A\).
   (4)

3. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 1.5 m and mass 8 kg . A particle of mass \(m \mathrm {~kg}\) is attached to the rod at \(B\). The rod is supported at the point \(C\), where \(A C = 0.9 \mathrm {~m}\), and the system is in equilibrium with \(A B\) horizontal, as shown in Figure 2.

1. Show that \(m = 2\). A particle of mass 5 kg is now attached to the rod at \(A\) and the support is moved from \(C\) to a point \(D\) of the rod. The system, including both particles, is again in equilibrium with \(A B\) horizontal.
2. Find the distance \(A D\).

3. A plank \(A B\) has length 6 m and mass 30 kg . The point \(C\) is on the plank with \(C B = 2 \mathrm {~m}\). The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg , stand on the plank. One person stands at the point \(P\) of the plank, where \(A P = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(A Q = 2 x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.

1. Find the value of \(x\).
2. State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

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 \(x\).

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 \(A E\).

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 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 \(A Q\).

2. \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 \(A Q\).
   . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_190_476_964_1905} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
   1. Find the size of the angle \(\alpha\).
      (6 marks)
   2. Find the value of \(T\). You must ensure that your answers to parts of questions are clearly labelled.
      You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.

3 The diagram shows a uniform rod, \(A B\), of mass 10 kg and length 5 metres. The rod is held in equilibrium in a horizontal position, by a support at \(C\) and a light vertical rope attached to \(A\), where \(A C\) is 2 metres. \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_237_680_479_648}

1. Draw and label a diagram to show the forces acting on the rod.
2. Show that the tension in the rope is 24.5 N .
3. A package of mass \(m \mathrm {~kg}\) is suspended from \(B\). The tension in the rope has to be doubled to maintain equilibrium.
   1. Find \(m\).
   2. Find the magnitude of the force exerted on the rod by the support.
4. Explain how you have used the fact that the rod is uniform in your solution.

4 Ken is trying to cross a river of width 4 m . He has a uniform plank, \(A B\), of length 8 m and mass 17 kg . The ground on both edges of the river bank is horizontal. The plank rests at two points, \(C\) and \(D\), on fixed supports which are on opposite sides of the river. The plank is at right angles to both river banks and is horizontal. The distance \(A C\) is 1 m , and the point \(C\) is at a horizontal distance of 0.6 m from the river bank. Ken, who has mass 65 kg , stands on the plank directly above the middle of the river, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-3_468_1086_1710_479}

1. Draw a diagram to show the forces acting on the plank.
2. Given that the reaction on the plank at the point \(D\) is \(44 g \mathrm {~N}\), find the horizontal distance of the point \(D\) from the nearest river bank.
3. State how you have used the fact that the plank is uniform in your solution.

6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h] \end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.

1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
   \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
   \section*{END OF QUESTION PAPER}

7. As part of a design for a new building, an architect wants to support a wooden beam in a horizontal position. The beam is suspended using a vertical steel cable and a smooth fixed support on its underside. The diagram below shows the architect's diagram and the adjacent table shows the categories of steel cable available. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-18_504_1699_559_191} You may use the following modelling assumptions.

• The wooden beam is a rigid uniform rod of mass 100 kg .
• The force exerted on the beam by the support is vertical.
• The steel cable is inextensible.

\section*{SAFETY REQUIREMENT} Both the steel cable and the support must be capable of withstanding forces of at least four times those present in the architect's diagram above. The wooden beam is held in horizontal equilibrium.
[0pt]

1. 1. Given that the support is capable of withstanding loads of up to 2000 N , show that the force exerted on the beam by the support satisfies the safety requirement. [3]
   2. Determine which categories of steel cable in the table opposite could meet the safety requirements.
2. State how you have used the modelling assumption that the beam is a uniform rod. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

3 A uniform plank, of length 8 metres, has mass 30 kg . The plank is supported in equilibrium in a horizontal position by two smooth supports at the points \(A\) and \(B\), as shown in the diagram. A block, of mass 20 kg , is placed on the plank at point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-3_193_1216_477_404}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the magnitude of the force exerted on the plank by the support at \(B\) is \(19.2 g\) newtons.
3. Find the magnitude of the force exerted on the plank by the support at \(A\).
4. Explain how you have used the fact that the plank is uniform in your solution.

16 A straight uniform rod, \(A B\), has length 6 m and mass 0.2 kg A particle of weight \(w\) newtons is fixed at \(A\).
A second particle of weight \(3 w\) newtons is fixed at \(B\).
The rod is suspended by a string from a point \(x\) metres from \(B\).
The rod rests in equilibrium with \(A B\) horizontal and the string hanging vertically as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-24_410_1148_767_445} Show that $$x = \frac { 3 w + 0.3 g } { 2 w + 0.1 g }$$ \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-25_2488_1716_219_153}

14 A \(\pounds 2\) coin has a diameter of 28 mm and a mass of 12 grams. A uniform rod \(A B\) of length 160 mm and a fixed load of mass \(m\) grams are used to check that a \(\pounds 2\) coin has the correct mass. The rod rests with its midpoint on a support.
A \(\pounds 2\) coin is placed face down on the rod with part of its curved edge directly above \(A\). The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-22_195_766_854_639} 14

1. Given that the rod is horizontal and rests in equilibrium, find \(m\).
   14
2. State an assumption you have made about the \(\pounds 2\) coin to answer part (a).

17 A uniform plank \(P Q\), of length 7 metres, lies horizontally at rest, in equilibrium, on two fixed supports at points \(X\) and \(Y\) The distance \(P X\) is 1.4 metres and the distance \(Q Y\) is 2 metres as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_56_689_534_762} \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_225_830_607_694} 17

1. The reaction force on the plank at \(X\) is \(4 g\) newtons.
   17
   1. (i) Show that the mass of the plank is 9.6 kilograms.
      17
   2. (ii) Find the reaction force, in terms of \(g\), on the plank at \(Y\) 17
   3. The support at \(Y\) is moved so that the distance \(Q Y = 1.4\) metres. The plank remains horizontally at rest in equilibrium.
      It is claimed that the reaction force at \(Y\) remains unchanged.
      Explain, with a reason, whether this claim is correct.

\includegraphics{figure_5} A uniform rod AB has length \(2\) m and weight \(20\) N. The rod rests horizontally in equilibrium on two supports at points C and D, where AC = \(0.4\) m and BD = \(0.6\) m.

1. Find the reaction at each support. [4]
2. State what happens if the support at D is removed. [2]

\includegraphics{figure_1} A metal girder \(AB\), of weight 1080 N and length 6 m, rests in equilibrium in a horizontal position on two supports, one at \(C\) and one at \(D\), where \(AC = 0.5\) m and \(BD = 2\) m, as shown in Figure 1. A boy of weight 400 N stands on the girder at \(B\) and the girder remains horizontal and in equilibrium. The boy is modelled as a particle and the girder is modelled as a uniform rod.

1. Find
   1. the magnitude of the reaction on the girder at \(C\),
   2. the magnitude of the reaction on the girder at \(D\).
   [6]

The boy now stands at a point \(E\) on the girder, where \(AE = x\) metres, and the girder remains horizontal and in equilibrium. Given that the magnitude of the reaction on the girder at \(D\) is now 520 N greater than the magnitude of the reaction on the girder at \(C\),

1. find the value of \(x\). [5]

A beam \(AB\) has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 1\) m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\). [7]

\includegraphics{figure_3} A uniform rod \(AB\) has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points \(C\) and \(D\), where \(AC = 0.5\) m and \(AD = 2\) m, as shown in Fig. 3. A particle of weight \(W\) newtons is attached to the rod at a point \(E\) where \(AE = x\) metres. The rod remains in equilibrium and the magnitude of the reaction at \(C\) is now twice the magnitude of the reaction at \(D\).

1. Show that \(W = \frac{60}{1-x}\). [8]
2. Hence deduce the range of possible values of \(x\). [2]

\includegraphics{figure_1} A lever consists of a uniform steel rod \(AB\), of weight 100 N and length 2 m, which rests on a small smooth pivot at a point \(C\) of the rod. A load of weight 2200 N is suspended from the end \(B\) of the rod by a rope. The lever is held in equilibrium in a horizontal position by a vertical force applied at the end \(A\), as shown in Fig. 1. The rope is modelled as a light string. Given that \(BC = 0.2\) m,

1. find the magnitude of the force applied at \(A\). [4]

The position of the pivot is changed so that the rod remains in equilibrium when the force at \(A\) has magnitude 1200 N.

1. Find, to the nearest cm, the new distance of the pivot from \(B\). [5]

A steel girder \(AB\), of mass 200 kg and length 12 m, rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(AC = 2\) m and \(DB = 2\) m. A man of mass 80 kg stands on the girder at the point \(P\), where \(AP = 4\) m, as shown in Figure 1. The man is modelled as a particle and the girder is modelled as a uniform rod.

1. Find the magnitude of the reaction on the girder at the support at \(C\). [3]

The support at \(D\) is now moved to the point \(X\) on the girder, where \(XB = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find

1. the magnitude of the reaction at the support at \(X\), [2]
2. the value of \(x\). [4]

\includegraphics{figure_3} A uniform beam \(AB\) has mass 12 kg and length 3 m. The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(BC = 1\) m, as shown in Figure 3. The beam is modelled as a uniform rod.

1. Find the reaction on the beam at \(C\). [3]

A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.

1. Find the distance \(AD\). [7]

A plank \(PQR\), of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at \(P\) and \(Q\), where \(PQ = 6\) m. A child of mass 40 kg stands on the plank at a distance of 2 m from \(P\) and a block of mass \(M\) kg is placed on the plank at the end \(R\). The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at \(P\) is equal to the force exerted on the plank by the support at \(Q\). By modelling the plank as a uniform rod, and the child and the block as particles,

1. 1. find the magnitude of the force exerted on the plank by the support at \(P\),
   2. find the value of \(M\). [10]
2. State how, in your calculations, you have used the fact that the child and the block can be modelled as particles. [1]