6.04e Rigid body equilibrium: coplanar forces

A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]

A uniform lamina is in the form of a trapezium \(ABCD\), as shown. \(AB\) and \(DC\) are perpendicular to \(BC\). \(AB = 17\) cm, \(BC = 21\) cm and \(CD = 8\) cm. \includegraphics{figure_7}

1. Find the distances of the centre of mass of the lamina from
   1. \(AB\),
   2. \(BC\). [8 marks]

The lamina is freely suspended from \(C\) and rests in equilibrium.

1. Find the angle between \(CD\) and the vertical. [3 marks]

A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]

A rectangular piece of cardboard \(ABCD\), measuring \(30\) cm by \(12\) cm, has a semicircle of radius \(5\) cm removed from it as shown. \includegraphics{figure_6}

1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(AB\) and from \(BC\). [7 marks]

The remaining cardboard is suspended from \(A\) and hangs in equilibrium.

1. Find the angle made by \(AB\) with the vertical. [4 marks]

\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCDE\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm.

1. Find the distance of the centre of mass of the lamina from \(ED\). [4 marks]

The lamina has mass \(m\). A particle \(P\) is attached to the lamina at \(B\). The lamina is then suspended freely from \(A\) and hangs in equilibrium with \(AD\) vertical.

1. Find, in terms of \(m\), the mass of \(P\). [5 marks]

\(PQR\) is a triangular lamina with \(PQ = 18\) cm, \(QR = 24\) cm and \(PR = 30\) cm.

1. Verify that angle \(PQR\) is a right angle and find the distances of the centre of mass of the lamina from
   1. \(PQ\),
   2. \(QR\).
   [5 marks]

\includegraphics{figure_6} The lamina is held in a vertical plane and placed on a line of greatest slope of a rough plane inclined at an angle \(\theta\) to the horizontal, as shown.

1. Find the largest value of \(\theta\) for which equilibrium will not be broken by toppling. [4 marks]

A uniform solid cone has vertical height 20 cm and base radius \(r\) cm. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is \(24°\) (see diagram). \includegraphics{figure_1}

1. Find \(r\), correct to 1 decimal place. [4]

A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of \(24°\).

1. State, with justification, whether this cone will topple. [1]

\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]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).

1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]

The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.

1. Find the angle between \(CD\) and the vertical. [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]

You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. \includegraphics{figure_4_1} Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is \(h\) cm, where \(h\) can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres. \includegraphics{figure_4_2}

1. Write down the coordinates of the centre of mass of the triangle OAB. [1]
2. Show that the centre of mass of the region OABCD is \(\left(\frac{12-h^2}{2(h+3)}, 2.5\right)\). [6]

The \(x\)-axis is horizontal. The prism is placed on a horizontal plane in the position shown in Fig. 4.2.

1. Find the values of \(h\) for which the prism would topple. [3]

The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.

1. Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
2. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]

\includegraphics{figure_4_3}

1. Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from the centre \(O\) of the plane face. [7 marks]

The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{10}\). \includegraphics{figure_7} \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying special attention to their lines of action. [3 marks] \item Given that the plane face containing the diameter \(AB\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac{4}{5}\). [6 marks] \end{enumerate]

A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac{a}{2}\) from the centre and parallel to a diameter.

1. Show that the centre of gravity of the smaller cap from its plane face is \(\frac{7a}{40}\). [9 marks]

This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.

1. If the maximum value of \(\theta\) for which this is possible without the cap turning over is 30°, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical. [6 marks]

A container consists of two sections made from the same material: a hollow portion formed by removing a cone (shaded in the figure) from a solid cylinder of radius \(r\) and height \(h\), and a solid hemisphere of radius \(r\). The vertex of the removed cone coincides with the centre \(O\) of the horizontal plane face of the hemisphere. \(CD\) is a diameter of this plane face. \includegraphics{figure_7}

1. Show that the distance of the centre of mass of the container from the plane face of the hemisphere is \(\left|\frac{3}{8}(h-r)\right|\). Explain why the modulus sign is necessary. [9 marks]
2. Find the ratio \(h : r\) in each of the following cases:
   1. When the container is suspended from the point \(C\), the angle made by \(CD\) with the vertical is equal to the angle which \(CD\) would make with the vertical if the hemisphere alone were suspended from \(C\). [4 marks]
   2. The container is able to stand without toppling in any position when it is placed with the surface of the hemispherical part in contact with a smooth horizontal table. [3 marks]

1. Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3h}{4}\) from the vertex. [7 marks]

An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(kh\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face of the chocolate. \includegraphics{figure_3}

1. Show that \(k = \frac{8h}{15}\). [3 marks]

In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which the cone does not topple is \(45°\), \includegraphics{figure_4}

1. find the radius of the base of the cone in terms of \(h\). [4 marks]

A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown. \includegraphics{figure_4}

1. Show by integration that the centre of mass of the lamina is at a distance \(\frac{\pi^2 - 4}{2\pi}\) from the \(y\)-axis. [9 marks]

Given that the centre of mass is at a distance 0·75 units from the \(x\)-axis, and that \(P\) is the point \((0, 2)\) and \(O\) is the origin \((0, 0)\),

1. find, to the nearest degree, the angle between the line \(OP\) and the vertical when the lamina is freely suspended from \(P\). [3 marks]

\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_3} A uniform rod \(AB\), of mass \(m\) and length \(2a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(AC = 4a\). A light elastic string, of natural length \(2a\) and modulus of elasticity \(\frac{1}{4}mg\), joins \(B\) to \(C\). The rod \(AB\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.

1. Show that the potential energy of the system is $$-mga[\cos \theta + \sqrt{(5 - 4 \cos \theta)}] + \text{constant}.$$ [9]
2. Hence determine the values of \(\theta\) for which the system is in equilibrium. [6]

\includegraphics{figure_1} Figure 1 shows a uniform rod \(AB\), of mass \(m\) and length \(4a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{1}{4}mg\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle BAC = 2\theta\), where \(\theta < \frac{\pi}{4}\). Given that \(AC\) is always horizontal,

1. show that the potential energy of the system is $$\frac{mga}{8}(16\sin 2\theta + 3\cot^2 \theta - 6\cot \theta) + \text{constant}.$$ [7]
2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\). [6]
3. Determine whether this position of equilibrium is stable or unstable. [3]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), each of mass \(2m\) and length \(2L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A\), \(B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(PAB\) and \(PAC\) are each equal to \(\theta\) (\(\theta > 0\)), as shown in Fig. 2.

1. Find the length of \(AP\) in terms of \(s\), \(L\) and \(\theta\). [2]
2. Show that the potential energy \(V\) of the system is given by $$V = 2mgL(3\cos\theta + \sin\theta) + \text{constant}.$$ [4]
3. Hence find the value of \(\theta\) for which the system is in equilibrium. [4]
4. Determine whether this position of equilibrium is stable or unstable. [4]