6.04e Rigid body equilibrium: coplanar forces

4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}

1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
   Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
   (4 marks)

3 A uniform plank, of length 8 metres, has mass 30 kg . The plank is supported in equilibrium in a horizontal position by two smooth supports at the points \(A\) and \(B\), as shown in the diagram. A block, of mass 20 kg , is placed on the plank at point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-3_193_1216_477_404}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the magnitude of the force exerted on the plank by the support at \(B\) is \(19.2 g\) newtons.
3. Find the magnitude of the force exerted on the plank by the support at \(A\).
4. Explain how you have used the fact that the plank is uniform in your solution.

2 A uniform plank, of length 6 metres, has mass 40 kg . The plank is held in equilibrium in a horizontal position by two vertical ropes attached to the plank at \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-2_323_1162_1464_440}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the tension in the rope attached to the plank at \(B\) is \(21 g \mathrm {~N}\).
3. Find the tension in the rope that is attached to the plank at \(A\).
4. State where in your solution you have used the fact that the plank is uniform.

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).

1. Find, in terms of \(V\), the speed of \(S\)
   1. immediately before the collision,
   2. immediately after the collision.
2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
   

5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5

1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
   5
3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5 (c) (i) Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5 (c) (ii) Find the range of possible values for the coefficient of friction between the cone and the board.

8 A ladder has length 4 metres and mass 20 kg The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall. In this position the ladder is on the point of slipping.
The top of the wall is 1.5 metres above the horizontal surface.
The angle between the ladder and the horizontal surface is \(\alpha\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593} The coefficient of friction between the ladder and the wall is 0.5
The coefficient of friction between the ladder and the ground is also 0.5
Show that $$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$ stating clearly any assumptions you make. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}

2 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(y = 6 + \sin x\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 2.1). The centre of mass of \(L\) is at the point \(( \bar { x } , \bar { y } )\). \begin{figure}[h] \end{figure}

1. Show that \(\bar { x } = 2.36\), correct to 3 significant figures.
2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P \mathrm {~N}\) acting at \(B\) perpendicular to the cover (see Fig. 2.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6bf2fa5-2f29-4632-b27d-ed8c5a0379cf-03_412_213_402_525} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.

4 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point \(( 0,0 )\), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h] \end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).

1. - Show that \(\bar { x } = 0.38\).
   A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi . C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_605_828_1525_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
2. Show that \(l = 0.3\).
3. Show that \(\mu \geqslant \frac { 14 } { 27 }\).

\(A C B\) is the diameter of a semi-circular lamina of radius \(2 a\) and centre \(C\). Another semi-circular lamina, having \(A C\) as its diameter, is added to form a uniform lamina, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}

1. 1. Show that the distance of the centre of mass of the lamina from \(A B\) is \(\frac { 28 } { 15 \pi } a\).
   2. Calculate the distance of the centre of mass of the lamina from a line drawn through \(A\) that is perpendicular to \(A B\).
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      □
      
   (b) Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at \(A\) and \(B\) so that \(A B\) is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at \(B\).
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) shown in Figure 2 is in the shape of an isosceles trapezium.

• \(B C\) is parallel to \(A D\) and angle \(B A D\) is equal to angle \(A D C\)
• \(B C = 5 a\) and \(A D = 7 a\)
• the perpendicular distance between \(B C\) and \(A D\) is \(3 a\)
• the distance of the centre of mass of \(A B C D\) from \(A D\) is \(d\)
  1. Show that \(d = \frac { 17 } { 12 } a\)

The uniform lamina \(P Q R S\) is a rectangle with \(P Q = 5 a\) and \(Q R = 9 a\).
The lamina \(A B C D\) in Figure 2 is used to cut a hole in \(P Q R S\) to form the template shown shaded in Figure 3. \begin{figure}[h] \end{figure}

The template is freely suspended from \(P\) and hangs in equilibrium with \(P S\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.

Find the value of \(\theta\)

10 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
3. Show that \(0.5 < \tan \alpha \leqslant 1\).
4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.

12 A uniform \(\operatorname { rod } A B\) has mass 5 kg and length 4 m .

1. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_529_540_995_840} The rod rests with \(A\) on a rough plane that makes an angle of \(60 ^ { \circ }\) to the horizontal. A string is attached to \(B\) and the rod is in equilibrium in the vertical plane containing the line of greatest slope of the plane, with the string vertical and \(A B\) perpendicular to the plane (see diagram). Find the magnitude of the frictional force at \(A\) and the tension in the string.
2. \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-5_323_637_1850_794} The rod now rests horizontally with \(A\) in contact with a rough plane that makes an angle of \(60 ^ { \circ }\) with the horizontal and \(B\) in contact with a rough plane that makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). The rod and the lines of greatest slope of the two planes are all in the same vertical plane. The coefficients of friction at \(A\) and \(B\) are \(\mu _ { A }\) and \(\mu _ { B }\) respectively. Friction is limiting at both \(A\) and \(B\), with \(A\) on the point of slipping downwards. Show that \(\mu _ { B } = \frac { 1 - \alpha \mu _ { A } } { \alpha + \mu _ { A } }\) where \(\alpha\) is an irrational number to be found.

9 The diagram shows a uniform \(\operatorname { rod } A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-5_661_655_390_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.

9 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-5_657_659_392_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.

\includegraphics{figure_2} A uniform rod \(AB\) of weight \(W\) rests in equilibrium with \(A\) in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude \(P\) acting at \(B\) in this vertical plane. The rod makes an angle of \(60°\) with the wall, and the force makes an angle of \(30°\) with the rod (see diagram). Find the value of \(P\). [3] Find also the set of possible values of the coefficient of friction between the rod and the wall. [4]

\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{1}{2}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]

\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{14}{5}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]

\includegraphics{figure_4} A uniform rod \(AB\) of length \(4a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac{5}{2}a\) has one end attached to the point \(C\) on the rod, where \(AC = \frac{3}{2}a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the rod \(AB\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\). [10]

\includegraphics{figure_5} Two uniform rods, \(AB\) and \(BC\), each have length \(2a\) and weight \(W\). They are smoothly jointed at \(B\), and \(A\) is attached to a smooth fixed pivot. A light inextensible string of length \((2\sqrt{2})a\) joins \(A\) to \(C\) so that angle \(ABC = 90°\). The system hangs in equilibrium, with \(AB\) making an angle \(\alpha\) with the vertical (see diagram). By taking moments about \(A\) for the system, or otherwise, show that \(\alpha = 18.4°\), correct to the nearest \(0.1°\). [3] Find the tension in the string in the form \(kW\), giving the value of \(k\) correct to 3 significant figures. [3] Find, in terms of \(W\), the magnitude of the force acting on the rod \(BC\) at \(B\). [6]

\includegraphics{figure_4} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(AC\) is inclined at \(2\theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that

1. the contact force acting on the rod at \(A\) has magnitude \(W\tan\theta\),
2. the contact force acting on the rod at \(B\) has magnitude \(\frac{W\cos 2\theta}{\cos\theta}\),
3. \(2r\cos 2\theta = a\cos\theta\).

[9]

\includegraphics{figure_4} A uniform rod \(AB\), of length \(l\) and mass \(m\), rests in equilibrium with its lower end \(A\) on a rough horizontal floor and the end \(B\) against a smooth vertical wall. The rod is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{4}{3}\), and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring \(CD\) which is in compression in a vertical line with its lower end \(D\) fixed on the floor. The upper end \(C\) is attached to the rod at a distance \(\frac{4l}{5}\) from \(B\) (see diagram). The coefficient of friction at \(A\) between the rod and the floor is \(\frac{1}{2}\) and the system is in limiting equilibrium.

1. Show that the normal reaction of the floor at \(A\) has magnitude \(\frac{1}{2}mg\) and find the force in the spring. [7]
2. Given that the modulus of elasticity of the spring is \(2mg\), find the natural length of the spring. [4]

\includegraphics{figure_2} A uniform solid cone has height 30 cm and base radius \(r\) cm. The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35°\), when the cone topples. The diagram shows a cross-section of the cone.

1. Find the value of \(r\). [3]
2. Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]

\includegraphics{figure_4} A uniform lamina of weight 15 N is in the form of a trapezium \(ABCD\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(AB\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30°\) upwards from the horizontal (see diagram). Find the tension in the string. [5]

\includegraphics{figure_6} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(AB\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]

\(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.

1. Find the speed with which \(P\) passes through \(M\). [6]

\includegraphics{figure_7} The diagram shows the cross-section \(OABCDE\) through the centre of mass of a uniform prism on a rough inclined plane. The portion \(ADEO\) is a rectangle in which \(AD = OE = 0.6\) m and \(DE = AO = 0.8\) m; the portion \(BCD\) is an isosceles triangle in which angle \(BCD\) is a right angle, and \(A\) is the mid-point of \(BD\). The plane is inclined at \(45°\) to the horizontal, \(BC\) lies along a line of greatest slope of the plane and \(DE\) is horizontal.

1. Calculate the distance of the centre of mass of the prism from \(BD\). [3]

The weight of the prism is \(21\) N, and it is held in equilibrium by a horizontal force of magnitude \(P\) N acting along \(ED\).

1. 1. Find the smallest value of \(P\) for which the prism does not topple. [2]
   2. It is given that the prism is about to slip for this smallest value of \(P\). Calculate the coefficient of friction between the prism and the plane. [3]

The value of \(P\) is gradually increased until the prism ceases to be in equilibrium.

1. Show that the prism topples before it begins to slide, stating the value of \(P\) at which equilibrium is broken. [5]