Beam on point of tilting

1 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_182_843_264_651} A uniform rigid plank has mass 10 kg and length 4 m . The plank has 0.9 m of its length in contact with a horizontal platform. A man \(M\) of mass 75 kg stands on the end of the plank which is in contact with the platform. A child \(C\) of mass 25 kg walks on to the overhanging part of the plank (see diagram). Find the distance between the man and the child when the plank is on the point of tilting.

4. \begin{figure}[h] \end{figure} A non-uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(5 d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = D B = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\frac { 5 } { 2 } m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).

1. Show that \(G D = \frac { 5 } { 2 } d\). The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
2. Find the magnitude of the normal reaction between the support at \(D\) and the rod.

5. \begin{figure}[h] \end{figure} A large \(\log A B\) is 6 m long. It rests in a horizontal position on two smooth supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(B D = 1 \mathrm {~m}\), as shown in Figure 4. David needs an estimate of the weight of the log, but the log is too heavy to lift off both supports. When David applies a force of magnitude 1500 N vertically upwards to the \(\log\) at \(A\), the \(\log\) is about to tilt about \(D\).

1. State the value of the reaction on the \(\log\) at \(C\) for this case. David initially models the log as uniform rod. Using this model,
2. estimate the weight of the log The shape of the log convinces David that his initial modelling assumption is too simple. He removes the force at \(A\) and applies a force acting vertically upwards at \(B\). He finds that the log is about to tilt about \(C\) when this force has magnitude 1000 N. David now models the log as a non-uniform rod, with the distance of the centre of mass of the \(\log\) from \(C\) as \(x\) metres. Using this model, find
3. a new estimate for the weight of the log,
4. the value of \(x\).
5. State how you have used the modeling assumption that the log is a rod.

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 \(A D\).

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 \(M\).

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 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 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .

1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.

9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find

1. the magnitude of the reaction of the support on the plank at \(D\),
2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
3. Find the weight of the stone block.
4. Explain the limitation of modelling
   1. the stone block as a particle,
   2. the plank as a rigid rod.

4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h] \end{figure}

6 A uniform ruler AB has mass 28 g and length 30 cm . As shown in Fig. 6, the ruler is placed on a horizontal table so that it overhangs a point C at the edge of the table by 25 cm . A downward force of \(F \mathrm {~N}\) is applied at A . This force just holds the ruler in equilibrium so that the contact force between the table and the ruler acts through C . \begin{figure}[h] \end{figure}

1. Complete the force diagram in the Printed Answer Booklet, labelling the forces and all relevant distances.
2. Calculate the value of \(F\). Answer all the questions.
   Section B (78 marks)

4. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of mass 50 kg and length 5 m which overhangs a river by 2 m . When a boy of mass 20 kg stands at \(A\), his sister can walk to within 0.3 m of \(B\), at which point the plank is in limiting equilibrium.

1. What is the mass of the girl?
2. Find the smallest extra weight which must be placed at \(A\) to enable the girl to walk right to the end \(B\).
3. How have you used the fact that the plank is uniform?

6. \begin{figure}[h] \end{figure} Figure 2 shows a picnic bench of mass 20 kg which consists of a horizontal plank of wood of length 2 m resting on two supports, each of which is 0.6 m from the centre of the plank. Luigi sits on the bench at its midpoint and his mother Maria sits at one end. Their masses are 40 kg and 75 kg respectively. By modelling the bench as a uniform rod and Luigi and Maria as particles,

1. find the reaction at each of the two supports. Luigi moves to sit closer to his mother.
2. Find how close Luigi can get to his mother before the reaction at one of the supports becomes zero.
3. Explain the significance of a zero reaction at one of the supports.

6. \begin{figure}[h] \end{figure} Figure 2 shows a bench of length 3 m being used in a gymnasium.
The bench rests horizontally on two identical supports which are 2.2 m apart and equidistant from the middle of the bench.

1. Explain why it is reasonable to model the bench as a uniform rod. When a gymnast of mass 55 kg stands on the bench 0.1 m from one end, the bench is on the point of tilting.
2. Find the mass of the bench. The first gymnast dismounts and a second gymnast of mass 33 kg steps onto the bench at a distance of 0.4 m from its centre.
3. Show that the magnitudes of the reaction forces on the two supports are in the ratio \(5 : 3\).
   (6 marks)

2 A uniform beam, AB , is 6 m long and has a weight of 240 N .
Initially, the beam is in equilibrium on two supports at C and D, as shown in Fig. 2.1. The beam is horizontal. \begin{figure}[h] \end{figure}

1. Calculate the forces acting on the beam from the supports at C and D . A workman tries to move the beam by applying a force \(T \mathrm {~N}\) at A at \(40 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam remains in horizontal equilibrium but the reaction of support C on the beam is zero. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_318_691_1119_687} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. (A) Calculate the value of \(T\).
   (B) Explain why the support at D cannot be smooth. The beam is now supported by a light rope attached to the beam at A , with B on rough, horizontal ground. The rope is at \(90 ^ { \circ }\) to the beam and the beam is at \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 2.3. The tension in the rope is \(P \mathrm {~N}\). The beam is in equilibrium on the point of sliding. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_438_633_1909_708} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
3. (A) Show that \(P = 60 \sqrt { 3 }\) and hence, or otherwise, find the frictional force between the beam and the ground.
   (B) Calculate the coefficient of friction between the beam and the ground.

\includegraphics{figure_1} A diving board \(AB\) 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 \(AC = 0.6\) 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 \(C\).
   [6] 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]
3. Explain how you have used the fact that the diver is modelled as a particle. [1]

\includegraphics{figure_2} A uniform plank \(AB\) has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1\) m and \(CD = x\) m, as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N. Modelling the plank as a rod,

1. show that \(x = 0.75\) [3]

A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find

1. the weight of the rock, [4]
2. the magnitude of the reaction of the support on the plank at \(D\). [2]
3. State how you have used the model of the rock as a particle. [1]

\includegraphics{figure_2} A plank \(AE\), of length \(6\) m and mass \(10\) kg, rests in a horizontal position on supports at \(B\) and \(D\), where \(AB = 1\) m and \(DE = 2\) m. A child of mass \(20\) kg stands at \(C\), the mid-point of \(BD\), as shown in Fig. 2. The child is modelled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate

1. the magnitude of the force exerted by the support on the plank at \(B\), [4]
2. the magnitude of the force exerted by the support on the plank at \(D\). [3]

The child now stands at a point \(F\) on the plank. The plank is in equilibrium and on the point of tilting about \(D\).

1. Calculate the distance \(DF\). [4]

\includegraphics{figure_2} A plank \(AB\) has length \(4\) m. It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass \(40\) kg and Jill has mass \(25\) kg. They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(BC = 1.6\) m, the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,

1. find the mass of the plank. [3]

They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).

1. Find the distance \(BC\) in this position. [5]
2. State how you have used the modelling assumptions that
   1. the plank is uniform,
   2. the plank is a rod,
   3. Jack and Jill are particles.
   [3]

\includegraphics{figure_2} A plank of wood \(AB\) has mass 10 kg and length 4 m. It rests in a horizontal position on two smooth supports. One support is at the end \(A\). The other is at the point \(C\), 0.4 m from \(B\), as shown in Figure 2. A girl of mass 30 kg stands at \(B\) with the plank in equilibrium. By modelling the plank as a uniform rod and the girl as a particle,

1. find the reaction on the plank at \(A\). [4]

The girl gets off the plank. A boulder of mass \(m\) kg is placed on the plank at \(A\) and a man of mass 80 kg stands on the plank at \(B\). The plank remains in equilibrium and is on the point of tilting about \(C\). By modelling the plank again as a uniform rod, and the man and the boulder as particles,

1. find the value of \(m\). [4]

A uniform plank \(XY\) has length 7 m and mass 2 kg. It is placed with the portion \(ZY\) in contact with a horizontal surface, where \(ZY = 2.8\) m. To prevent the plank from toppling, a stone is placed on the plank at \(Y\). \includegraphics{figure_2}

1. Find the smallest possible mass of the stone. [4 marks]
2. State, with a reason, whether your answer to part (a) would be greater or smaller if a shorter portion of the plank were in contact with the surface. [2 marks]

\includegraphics{figure_7} A barrier is modelled as a uniform rectangular plank of wood, \(ABCD\), rigidly joined to a uniform square metal plate, \(DEFG\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m. The metal plate has mass 80 kg and side 0.5 m. The plank and plate are joined in such a way that \(CDE\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(CH\) is 0.25 m.

1. Calculate the contact force at \(H\) when the barrier is in the closed position. [3]

In the open position, the centre of mass of the barrier is vertically above \(D\).

1. Calculate the angle between \(AB\) and the horizontal when the barrier is in the open position. [8]

A metal rod, of mass \(m\) kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, \(B\), extends 6 cm beyond the edge of the shelf, \(A\), as shown in the diagram below. \includegraphics{figure_14}

1. The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of \(AB\). Show that \(m = 0.21\) [3 marks]
2. The object of mass 0.28 kilograms is removed. A number, \(n\), of identical objects, each of mass 0.048 kg, are hung from the rod all at a distance of 1 cm from \(B\). Find the maximum value of \(n\) such that the rod remains horizontal. [4 marks]
3. State one assumption you have made about the rod. [1 mark]