3.04b Equilibrium: zero resultant moment and force

\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]

\includegraphics{figure_2} A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.

1. Show that the tension in the rope attached to the pole at \(C\) is \(\left(\frac{5}{6}W + \frac{100}{3}\right)\) N. [4]
2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). [3]

Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),

1. find the value of \(W\). [3]

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_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]

\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_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]

\includegraphics{figure_2} A beam \(AB\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(AP = 0.3\) m and \(BQ = 0.3\) m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(PX = x\) m, \(0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

1. Show that the tension in the rope attached to the beam at \(P\) is \((588 - 350x)\) N. [3]
2. Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\). [3]
3. Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]

Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),

1. find the value of \(x\). [3]

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]

\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 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 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 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 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]

\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]

\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\), of mass \(m\) and length \(2a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(AC = \frac{2}{3}a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(BD = 2a\).

1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac{1}{2}mg \sin \theta\) [3]
2. Find the magnitude of the normal reaction on the rod at \(B\). [5]

The rod is in limiting equilibrium when \(\tan \theta = \frac{4}{3}\).

1. Find the coefficient of friction between the rod and the ground. [3]

\includegraphics{figure_2} Figure 2 A uniform rod \(AB\) has length \(4a\) and weight \(W\). A particle of weight \(kW\), \(k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(AC = 3a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{1}{3}\). The peg is perpendicular to the vertical plane containing \(AB\).

1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod. [1]
2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac{\sqrt{10}}{5}W(1 + 2k)$$ [4]

The coefficient of friction between the rod and the ground is \(\frac{3}{4}\).

1. Show that for the rod to remain in equilibrium \(k \leq \frac{2}{11}\). [7]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(0.15\) and \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of \(30°\) with the wall. A man of mass \(5m\) stands on the ladder, which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(6a\). Find the value of \(k\). [9]

\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(θ\) to the horizontal, where \(\tan θ = \frac{5}{3}\). The pole is modelled as a uniform rod.

1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]

The vertical component of the force exerted by the wall on the pole is \(\frac{1}{2}W\). Find

1. x in terms of \(a\), [3]

1. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). The cable makes an angle \(β\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is \(\frac{1}{3}\). The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{3}{4}W\). [6]

1. Find the value of \(β\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

END

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(j\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(j\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]

1. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(α\) with the horizontal, where \(\tan α = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{4}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]