3.04a Calculate moments: about a point

\includegraphics{figure_6} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor arc \(AB\) and the lines \(AT\) and \(BT\). Angle \(AOB\) is \(2\theta\) radians.

1. In the case where the area of the sector \(AOB\) is the same as the area of the shaded region, show that \(\tan \theta = 2\theta\). [3]
2. In the case where \(r = 8\) cm and the length of the minor arc \(AB\) is 19.2 cm, find the area of the shaded region. [3]

\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{5}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]

\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{3}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]

\includegraphics{figure_4} A uniform rod \(AB\) of length \(3a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length \(4a\). The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5a\). The string and the rod make angles \(\theta\) and \(2\theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac{3}{5}\) and you may use the fact that \(\cos 2\theta = \frac{7}{25}\).

1. Find the tension in the string in terms of \(W\). [3]
2. Find the modulus of elasticity of the string in terms of \(W\). [4]
3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal. [3]

\includegraphics{figure_1} A uniform ladder \(AB\), of length \(3a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(AC = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(DC\) is in the same vertical plane as the ladder \(AB\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \alpha = 2\tan \theta\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\). [9]

A uniform rod \(AB\) of length \(4a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(AC = 3a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(ACD = 2\theta\). A particle of weight \(\frac{1}{4}W\) is attached to the rod at \(B\). It is given that \(\tan \theta = \frac{5}{12}\).

1. Show that the tension in the string is \(\frac{17}{12}W\). [4]
2. Find the magnitude and direction of the reaction at the hinge. [5]
3. Given that the natural length of the string is \(2a\), find its modulus of elasticity. [2]

\includegraphics{figure_4} A uniform rod \(AB\) of length \(2a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(AC = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45°\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\). \begin{enumerate}[label=(\roman*)] \item Find an expression for \(x\) in terms of \(a\) and \(\mu\). [5] \item Hence show that \(\mu \geqslant \frac{1}{3}\). [2] \item Given that \(x = \frac{5}{3}a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\). [4] \end{enumerate]

\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).

1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
2. Given that the lamina is about to slip, find the value of \(\mu\). [3]

\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_4} A uniform beam \(AB\) has length \(2 \text{ m}\) and weight \(70 \text{ N}\). The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point \(1.7 \text{ m}\) vertically above the hinge. The other end of the rope is attached to the beam at a point \(0.8 \text{ m}\) from \(A\). The rope is at right angles to \(AB\). The beam carries a load of weight \(220 \text{ N}\) at \(B\) (see diagram).

1. Find the tension in the rope. [3]
2. Find the direction of the force exerted on the beam at \(A\). [4]

\includegraphics{figure_4} A uniform rod \(AB\) has weight \(15\) N and length \(1.2\) m. The end \(A\) of the rod is in contact with a rough plane inclined at \(30°\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).

1. Show that the magnitude of the force applied at \(B\) is \(4.33\) N, correct to \(3\) significant figures. [3]
2. Find the magnitude of the frictional force exerted by the plane on the rod. [2]
3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane. [3]

\includegraphics{figure_1} A non-uniform rod \(AB\), of length 0.6 m and weight 9 N, has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30°\) to the horizontal. The tension in the string is \(T\) N (see diagram).

1. Calculate \(T\). [2]
2. Find the least possible value of the coefficient of friction at \(A\). [3]

\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\), [3]
2. the least possible value of the coefficient of friction at \(B\). [4]

\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_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]

\includegraphics{figure_1} A uniform rod \(AB\) has length \(2a\) and mass \(M\). The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at \(C\) and \(D\), where \(AC = \frac{2}{5}a\) and \(DB = \frac{3}{5}a\), as shown in Figure 1. A particle \(P\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the largest possible mass of the particle \(P\) [3] Given that the mass of \(P\) is \(\frac{1}{2}M\)
2. Find, in terms of \(M\) and \(g\), the tension in the string that is attached to the rod at \(C\). [3]

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_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_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]

\includegraphics{figure_3} A particle \(A\) of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground. Particle \(A\) is connected to a particle \(B\), of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from \(A\) to the pulley lies in a line of greatest slope of the wedge. The particle \(B\) hangs freely below the pulley, as shown in Fig. 3. The system is released from rest with the string taut. For the motion before \(A\) reaches the pulley and before \(B\) hits the ground, find

1. the tension in the string, [6]
2. the magnitude of the resultant force exerted by the string on the pulley. [3]

1. The string in this question is described as being 'light'.
   1. Write down what you understand by this description.
   2. State how you have used the fact that the string is light in your answer to part (a). [2]

\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_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_3} A fixed wedge has two plane faces, each inclined at \(30°\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3. The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\frac{1}{10}g\).

1. By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. [3]
2. By considering the motion of \(B\), find the value of \(\mu\). [8]
3. Find the resultant force exerted by the string on the pulley, giving its magnitude and direction. [3]

\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]