Uniform beam on two supports

\includegraphics{figure_5} A uniform rod \(AB\) has length 2 m and mass 50 kg. The rod is in equilibrium in a horizontal position, resting on two smooth supports at \(C\) and \(D\), where \(AC = 0.2\) metres and \(DB = x\) metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at \(D\) is twice the magnitude of the reaction on the rod at \(C\),

1. find the value of \(x\). [6]

The support at \(D\) is now moved to the point \(E\) on the rod, where \(EB = 0.4\) metres. A particle of mass \(m\) kg is placed on the rod at \(B\), and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at \(E\) is four times the magnitude of the reaction on the rod at \(C\),

1. find the value of \(m\). [7]

A plank of wood \(AB\), of mass 8 kg and length 6 m, rests on a support at \(P\), where \(AP = 4\) m. When particles of mass 1 kg and \(k\) kg are suspended from \(A\) and \(B\) respectively, the plank rests horizontally in equilibrium. Modelling the plank as a uniform rod, find

1. the value of \(k\), [3 marks]
2. the magnitude of the force exerted by the support on the plank at \(P\). [2 marks]

\(AB\) is a light rod. Forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\), of magnitudes \(3\) N, \(2\) N and \(6\) N respectively, act upwards at right angles to the rod in a vertical plane at points dividing \(AB\) in the ratio \(1:4:2:4\), as shown. \includegraphics{figure_4} A single force \(\mathbf{P}\) is applied downwards at the point \(C\) to keep the rod horizontal in equilibrium.

1. State the magnitude of \(\mathbf{P}\). [1 mark]
2. Show that \(AC:CB = 5:6\). [5 marks]

Two particles, of weights \(3\) N and \(k\) N, are now placed on the rod at \(A\) and \(B\) respectively, while the same upward forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\) act as before. It is found that a single downward force at the same point \(C\) as before keeps \(AB\) horizontal under gravity.

1. Find the value of \(k\). [6 marks]

A boy holds a 30 cm metal ruler between three fingers of one hand, pushing down with the middle finger and up with the other two, at the points marked 5 cm, 10 cm and \(x\) cm and exerting forces of magnitude 11 N, 18 N and 8 N respectively. The ruler is in equilibrium in this position. Modelling the ruler as a uniform rod, find \includegraphics{figure_1}

1. the mass of the ruler, in grams, \hfill [3 marks]
2. the value of \(x\). \hfill [3 marks]
3. State how you have used the modelling assumption that the ruler is a uniform rod. \hfill [1 mark]

\includegraphics{figure_1} Figure 1 shows a uniform plank \(AB\) of length 8 m and mass 30 kg. It is supported in a horizontal position by two pivots, one situated at \(A\) and the other 2 m from \(B\). A man whose mass is 80 kg is standing on the plank 2 m from \(A\) when his dog steps onto the plank at \(B\). Given that the plank remains in equilibrium and that the magnitude of the forces exerted by each of the pivots on the plank are equal,

1. calculate the magnitude of the force exerted on the plank by the pivot at \(A\), [5 marks]
2. find the dog's mass. [3 marks]

If the dog was heavier and the plank was on the point of tilting,

1. explain how the force exerted on the plank by each of the pivots would be changed. [2 marks]

Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m. The beam is freely pivoted on a fixed support at D and is supported at C. The distance CD is 2.75 m. \includegraphics{figure_4} The beam is horizontal and in equilibrium.

1. Show that the anticlockwise moment of the weight of the beam about D is 1100 N m. Find the value of the normal reaction on the beam of the support at C. [6]

The support at C is removed and spheres at P and Q are suspended from the beam by light strings attached to the points C and R. The sphere at P has weight 440 N and the sphere at Q has weight \(W\) N. The point R of the beam is 1.5 m from D. This situation is shown in Fig. 4.2.

1. The beam is horizontal and in equilibrium. Show that \(W = 1540\). [3]

The sphere at P is changed for a lighter one with weight 400 N. The sphere at Q is unchanged. The beam is now held in equilibrium at an angle of 20° to the horizontal by means of a light rope attached to the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3. \includegraphics{figure_5}

1. Calculate the tension in the rope when it is
   1. at 90° to the beam, [6]
   2. horizontal. [3]

A uniform rod, AB, has length 4 metres. The rod is resting on a support at its midpoint C. A particle of mass 4 kg is placed 0.6 metres to the left of C. Another particle of mass 1.5 kg is placed \(x\) metres to the right of C, as shown. \includegraphics{figure_3} The rod is balanced in equilibrium at C. Find \(x\). Circle your answer. [1 mark] 1.8 m 1.5 m 1.75 m 1.6 m

A uniform rod, \(AB\), has length \(7\) metres and mass \(4\) kilograms. The rod rests on a single fixed pivot point, \(C\), where \(AC = 2\) metres. A particle of weight \(W\) newtons is fixed at \(A\), as shown in the diagram. \includegraphics{figure_13} The system is in equilibrium with the rod resting horizontally.

1. Find \(W\), giving your answer in terms of \(g\). [2 marks]
2. Explain how you have used the fact that the rod is uniform in part (a). [1 mark]

A uniform rod is resting on two fixed supports at points \(A\) and \(B\). \(A\) lies at a distance \(x\) metres from one end of the rod. \(B\) lies at a distance \((x + 0.1)\) metres from the other end of the rod. The rod has length \(2L\) metres and mass \(m\) kilograms. The rod lies horizontally in equilibrium as shown in the diagram below. \includegraphics{figure_17} The reaction force of the support on the rod at \(B\) is twice the reaction force of the support on the rod at \(A\). Show that $$L - x = k$$ where \(k\) is a constant to be found. [4 marks]

A uniform rod, \(AB\), has length 3 metres and mass 24 kg. A particle of mass \(M\) kg is attached to the rod at \(A\). The rod is balanced in equilibrium on a support at \(C\), which is 0.8 metres from \(A\). \includegraphics{figure_11} Find the value of \(M\). [2 marks]

The diagram shows a uniform plank \(AB\) of length 4 m supported in horizontal equilibrium by means of a central pivot. On the plank there are three objects of masses 8 kg, 2 kg and 15 kg placed in positions \(C\), \(D\) and \(E\) respectively. The distance \(AC\) is \(0 \cdot 6\) m and the distance \(AE\) is \(2 \cdot 8\) m. \includegraphics{figure_3} Find the distance \(AD\). [4]