3.04b Equilibrium: zero resultant moment and force

A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}

1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 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]

\includegraphics{figure_1} Figure 1 shows two window cleaners, Alan and Baber, of mass 60 kg and 100 kg respectively standing on a platform \(PQ\) of length 3 metres and mass 20 kg. The platform is suspended by two vertical cables attached to the ends \(P\) and \(Q\). Alan is standing at the point \(A\), 1.25 metres from \(P\), Baber is standing at the point \(B\) and the tension in the cable at \(P\) is twice the tension in the cable at \(Q\). Modelling the platform as a uniform rod and Alan and Baber as particles,

1. find the tension in the cable at \(P\), [2 marks]
2. find the distance \(BP\). [5 marks]
3. State how you have used the modelling assumptions that
   1. the platform is uniform,
   2. the platform is a rod.
   [2 marks]

A non-uniform plank \(AB\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is 35g N.

1. Suggest a suitable model for
   1. the plank, [2 marks]
   2. the woman.
2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\). [2 marks]
3. Explain briefly, in the context of the problem, the term 'non-uniform'. [2 marks]
4. Find the distance of the centre of mass of the plank from \(A\). [4 marks]

\includegraphics{figure_1} Figure 1 shows an aerial view of a revolving door consisting of 4 panels, each of width 1.2 m and set at 90° intervals, which are free to rotate about a fixed central column, \(O\). The revolving door is situated outside a lecture theatre and four students are trying to push the door. Two of the students are pushing panels \(OA\) and \(OD\) clockwise (as viewed from above) with horizontal forces of 70 N and 90 N respectively, whilst the other two are pushing panels \(OB\) and \(OC\) anti-clockwise with horizontal forces of 80 N and 60 N respectively.

1. Calculate the total moment about \(O\) when the four students are pushing the panels at their outer edge, 1.2 m from \(O\). [3 marks]

The student at \(C\) moves her hand 0.2 m closer to \(O\) and the student at \(D\) moves his hand \(x\) m closer to \(O\). Given that the students all push in the same directions and with the same forces as in part (a), and that the door is in equilibrium,

1. Find the value of \(x\). [5 marks]

\includegraphics{figure_1} Figure 1 shows a plank \(AB\) of mass 40 kg and length 6 m, which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end \(A\) than at end \(B\). A woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\).

1. Suggest suitable modelling assumptions which can be made about
   1. the plank,
   2. the woman. [3 marks]
   Given that the reactions at each support are of equal magnitude,
2. find the magnitude of the reaction on the support at \(A\), [2 marks]
3. calculate the distance of the centre of mass of the plank from \(A\). [4 marks]

A uniform ladder \(AB\), of length 6 metres and mass 22 kg, rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(AC\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60°\). A man, of mass 88 kg, is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(AD\) is 4 metres. The ladder is on the point of slipping. \includegraphics{figure_7}

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find the coefficient of friction between the ladder and the horizontal ground. [6 marks]

A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]

\includegraphics{figure_7} A uniform ladder \(AB\), of mass \(m\) kg and length \(2a\) m, rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{3}{4}\).

1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac{\sqrt{13}}{3}mg\). [7 marks]
2. Find, to the nearest degree, the direction of this resultant force at \(B\). [3 marks]

The peg will break when the horizontal force acting on it exceeds \(2mg\) N. A painter of mass \(6m\) kg starts to climb the ladder from \(B\).

1. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb. [7 marks]

\includegraphics{figure_5} A uniform rod \(AB\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).

1. Show that the tension in the string is 4.5 N. [4]
2. Find the magnitude and direction of the force acting on the rod at \(A\). [6]

\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find

1. the force acting on the rod at \(P\), [3]
2. the coefficient of friction between the ground and the rod. [5]

A non-uniform beam \(AB\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(AG = 2.5\) m. The beam is freely suspended from its end \(A\) and is held in a horizontal position by means of a wire attached to the end \(B\). The wire makes an angle of \(20°\) with the vertical and the tension is \(T\) N (see diagram).

1. Calculate \(T\). [3]
2. Calculate the magnitude and the direction of the force acting on the beam at \(A\). [7]

\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).

1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]

\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).

1. Calculate the tension in the string. [3]
2. Find the direction of the force acting on the lamina at \(A\). [4]

A uniform ladder \(AB\), of weight \(W\) and length \(2a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{12}{13}\). A man of weight \(6W\) is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{5W}{24}\left(1 + \frac{6x}{a}\right)\). [5]

The coefficient of friction between the ladder and the ground is \(\frac{1}{3}\).

1. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. [3]

The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight \(6W\) can now stand at the top of the ladder \(B\) without the ladder slipping.

1. Find the least possible value of \(\tan \alpha\). [3]

\includegraphics{figure_2} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD, BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.] The rods AB, BC and DE are horizontal. The rods are freely pin-jointed to each other at A, B, C, D and E. The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD. The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(LN\) at C; the normal reaction force \(RN\) of the plane on the framework at D; the horizontal and vertical forces \(XN\) and \(YN\), respectively, acting at A.

1. Write down equations for the horizontal and vertical equilibrium of the framework. [3]
2. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt{3}L\) and \(Y = 0\). [4]
3. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods. [2]
4. Show that the internal force in the rod AD is zero. [2]
5. Find the forces internal to AB, CE and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.] [7]
6. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust. [2]

A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}

1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]

The base OABC is added to the vertical faces.

1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]

The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.

1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]

[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.

1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
2. Calculate the value of \(P\). [2]

A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}

1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]

The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.

1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]

The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.

1. Calculate the value of \(P\). [5]

The coefficient of friction between the fire-screen and the floor is \(\mu\).

1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]

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]

\includegraphics{figure_3} Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal, light, rigid rods AB, AD, BC, BD and CD so that ABD and BCD are equilateral triangles of side 2 m. AD and BC are horizontal. The rods are freely pin-jointed to each other at A, B, C and D. The pin-joint at A is fixed to a wall and the pin-joint at B rests on a smooth horizontal support. Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and a horizontal force of 50 N applied at D; the normal reaction of the support on the framework at B is \(R\) N; horizontal and vertical forces \(X\) N and \(Y\) N act at A.

1. Write down equations for the horizontal and vertical equilibrium of the framework. [2]
2. Show that \(R = 135\) and \(Y = 90\). [3]
3. On the diagram in your printed answer book, show the forces internal to the rods acting on the pin-joints. [2]
4. Calculate the forces internal to the five rods, stating whether each rod is in tension or compression (thrust). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (iii).] [10]
5. Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system remains in equilibrium in the same position. By considering the equilibrium at C, show that the forces in rods CD and BC are not changed. [2]

\includegraphics{figure_2} Two uniform rods, \(AB\) and \(BC\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is \(1.5\) m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is \(3\) m (see diagram). The weight of \(AB\) is \(80\) N and the frictional force acting on \(AB\) at \(A\) is \(14\) N.

1. Write down the horizontal component of the force acting on \(AB\) at \(B\) and show that the vertical component of this force is \(33\) N upwards. [4]
2. Given that the force acting on \(BC\) at \(C\) has magnitude \(50\) N, find the weight of \(BC\). [4]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\) are of equal length and each has weight \(100\) N. The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(AB\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(AB\) (see diagram).

1. By taking moments about \(A\) for \(AB\), find the vertical component of the force on \(AB\) at \(B\). Hence find the vertical component of the contact force on \(BC\) at \(C\). [3]
2. Calculate the magnitude of the frictional force on \(BC\) at \(C\) and state its direction. [4]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), of lengths \(3\) m and \(4\) m respectively, have weights \(300\) N and \(400\) N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is \(2.4\) m above the surface (see diagram).

1. Show that the force exerted by the surface on \(AB\) is \(374\) N and find the force exerted by the surface on \(AC\). [4]
2. Find the tension in the string. [3]
3. Find the horizontal and vertical components of the force exerted on \(AB\) at \(A\) and state their directions. [3]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\), each of length \(2L\), are freely jointed at \(B\), and \(AB\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of rod \(BC\) is \(75\) N, and the tension in the string is \(50\) N (see diagram).

1. Show that \(\tan \beta = \frac{1}{3}\). [3]
2. Given that \(\tan \alpha = \frac{12}{5}\), find the weight of \(AB\). [5]

\includegraphics{figure_6} Two uniform rods \(AB\) and \(AC\) are freely jointed at \(A\). Rod \(AB\) is of length \(2l\) and weight \(W\); rod \(AC\) is of length \(4l\) and weight \(2W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(AB\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\), and \(AC\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac{1}{5}\) (see diagram). The force of the step acting on \(AB\) at \(B\) has vertical component \(R\) and horizontal component \(F\).

1. By taking moments about \(A\) for the rod \(AB\), find an equation relating \(W\), \(R\) and \(F\). [3]
2. Show that \(R = \frac{75}{68}W\), and find the vertical component of the force acting on \(AC\) at \(C\). [6]
3. The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction. [4]