Non-uniform beam on supports

6. \begin{figure}[h] \end{figure} A plank \(A B\) has mass 12 kg and length 2.4 m . A load of mass 8 kg is attached to the plank at the point \(C\), where \(A C = 0.8 \mathrm {~m}\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes, one attached at \(A\) and the other attached at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings.

1. Find the tension in the rope attached at \(B\). The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at \(A\) is 10 N greater than the tension in the rope attached at \(B\).
2. Find the distance of the centre of mass of the plank from \(A\).

2. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find

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

6. \begin{figure}[h] \end{figure} A non-uniform beam \(A D\) has weight \(W\) newtons and length 4 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(A B = 1 \mathrm {~m}\) and \(C D = 1 \mathrm {~m}\), as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.

1. Find the distance of the centre of mass of the beam from \(A\). A small load of weight \(k W\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. Find
2. an expression for the tension in the rope attached to \(B\), giving your answer in terms of \(k\) and \(W\),
3. the set of possible values of \(k\) for which both ropes remain taut.

6. A non-uniform plank \(A B\) has length 6 m and mass 30 kg . The plank rests in equilibrium in a horizontal position on supports at the points \(S\) and \(T\) of the plank where \(A S = 0.5 \mathrm {~m}\) and \(T B = 2 \mathrm {~m}\). When a block of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(S\). When the block is moved to \(B\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(T\). The distance of the centre of mass of the plank from \(A\) is \(d\) metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find

1. the value of \(d\),
2. the value of \(M\).
   

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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 \(A\).

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}

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 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 \(x\)

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 \(M\).

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 \(d\).

2. A plank of wood \(X Y\) has length \(5 a\) m and mass 5 kg . It rests on a support at \(Q\), where \(X Q = 3 a\) m . When a kitten of mass 8 kg sits on the plank at \(P\), where \(P Y = a \mathrm {~m}\), the plank just remains horizontal. By modelling the plank as a non-uniform rod and the kitten as a particle, find

1. the magnitude of the reaction at the support,
2. the distance from \(X\) to the centre of mass of the plank, in terms of \(a\).

2. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 6 m and mass 50 kg . The beam rests horizontally on two supports at \(C\) and \(D\), where \(A C = 0.9 \mathrm {~m}\) and \(D B = 1.8 \mathrm {~m}\). A child of mass 25 kg stands on the beam at \(E\), where \(A E = E B = 3 \mathrm {~m}\), as shown in Figure 1. The beam is in equilibrium.
The magnitude of the normal reaction between the beam and the support at \(C\) is \(R _ { C }\) newtons. The magnitude of the normal reaction between the beam and the support at \(D\) is \(R _ { D }\) newtons. The beam is modelled as a rod and the child is modelled as a particle.
The centre of mass of the beam is between \(C\) and \(D\) and is a distance \(x\) metres from \(D\).
Given that \(2 R _ { D } = 3 R _ { C }\)

1. show that \(x = 1.38\) The child remains at \(E\) and a block of mass \(M \mathrm {~kg}\) is placed on the beam at \(B\).
   The block is modelled as a particle.
   Given that the beam is on the point of tilting,
2. find the value of \(M\).

\includegraphics{figure_2} A non-uniform rod \(AB\) of length \(0.5 \text{ m}\) and weight \(8 \text{ N}\) is freely hinged to a fixed point at \(A\). The rod makes an angle of \(30°\) with the horizontal with \(B\) above the level of \(A\). The rod is held in equilibrium by a force of magnitude \(12 \text{ N}\) acting in the vertical plane containing the rod at an angle of \(30°\) to \(AB\) applied at \(B\) (see diagram). Find the distance of the centre of mass of the rod from \(A\). [3]

\includegraphics{figure_3} \(ABC\) is an object made from a uniform wire consisting of two straight portions \(AB\) and \(BC\), in which \(AB = a\), \(BC = x\) and angle \(ABC = 90°\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(AB\) and the horizontal is \(\theta\) (see diagram).

1. Show that \(x^2 \tan \theta - 2ax - a^2 = 0\). [3]
2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\). [2]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,

1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
2. State what is implied by the modelling assumption that the beam is uniform. [1]

David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,

1. find the distance of the centre of mass of the beam from \(C\). [4]

\includegraphics{figure_1} A uniform rod \(AB\) has length \(100 \text{ cm}\). Two light pans are suspended, one from each end of the rod, by two strings which are assumed to be light and inextensible. The system forms a balance with the rod resting horizontally on a smooth pivot, as shown in Fig. 1. A particle of weight \(16 \text{ N}\) is placed in the pan at \(A\) and a particle of weight \(5 \text{ N}\) is placed in the pan at \(B\). The rod rests horizontally in equilibrium when the pivot is at the point \(C\) on the rod, where \(AC = 30 \text{ cm}\).

1. Find the weight of the rod. [3]

The particle in the pan at \(A\) is replaced by a particle of weight \(3.5 \text{ N}\). The particle of weight \(5 \text{ N}\) remains in the pan at \(B\). The rod now rests horizontally in equilibrium when the pivot is moved to the point \(D\).

1. Find the distance \(AD\). [4]
2. Explain briefly where the assumption that the strings are light has been used in your answer to part (a). [1]

A beam \(AB\) has length 15 m. The beam rests horizontally in equilibrium on two smooth supports at the points \(P\) and \(Q\), where \(AP = 2\) m and \(QB = 3\) m. When a child of mass 50 kg stands on the beam at \(A\), the beam remains in equilibrium and is on the point of tilting about \(P\). When the same child of mass 50 kg stands on the beam at \(B\), the beam remains in equilibrium and is on the point of tilting about \(Q\). The child is modelled as a particle and the beam is modelled as a non-uniform rod.

1. 1. Find the mass of the beam.
   2. Find the distance of the centre of mass of the beam from \(A\). [8]

When the child stands at the point \(X\) on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,

1. find \(AX\). [6]

\includegraphics{figure_2} A non-uniform rod \(AB\) has length 5 m and weight 200 N. The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1.5\) m and \(DB = 1\) m, as shown in Fig. 2. The centre of mass of \(AB\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N.

1. Show that \(50x - W = 100\). [5]

The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N.

1. Obtain another equation connecting \(W\) and \(x\). [3]
2. Calculate the value of \(x\) and the value of \(W\). [4]

\includegraphics{figure_2} A non-uniform plank of wood \(AB\) has length 6 m and mass 90 kg. The plank is smoothly supported at its two ends \(A\) and \(B\), with \(A\) and \(B\) at the same horizontal level. A woman of mass 60 kg stands on the plank at the point \(C\), where \(AC = 2\) m, as shown in Fig. 2. The plank is in equilibrium and the magnitudes of the reactions on the plank at \(A\) and \(B\) are equal. The plank is modelled as a non-uniform rod and the woman as a particle. Find

1. the magnitude of the reaction on the plank at \(B\), [2]
2. the distance of the centre of mass of the plank from \(A\). [5]
3. State briefly how you have used the modelling assumption that
   1. the plank is a rod,
   2. the woman is a particle.
   [2]

An iron bar \(AB\), of length \(4\) m, is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(PB = 0.85\) m. The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(AG = 1.4\) m, and the wire is modelled as a light inextensible string. Given that the tension in the wire is \(12\) N, calculate

1. the weight of the bar, \hfill [4 marks]
2. the magnitude of the reaction on the bar at \(A\). \hfill [2 marks]
3. State briefly how you have used the given modelling assumption about the bar. \hfill [1 mark]