6.04e Rigid body equilibrium: coplanar forces

7 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). 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 \(C D E\) 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 \(C H\) is 0.25 m .

1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.

3 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).

1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
2. Calculate the tensions in the strings.

1 The region bounded by the \(x\)-axis, the curve \(\mathrm { y } = \sqrt { 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x - 20 }\) and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution \(R\). The centre of mass of \(R\) is the point \(G\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_569_463_507_280}

1. Explain why the \(y\)-coordinate of \(G\) is 0 .
2. Find the \(x\)-coordinate of \(G\). \(P\) is a point on the edge of the curved surface of \(R\) where \(x = 4 . R\) is freely suspended from \(P\) and hangs in equilibrium.
3. Find the angle between the axis of symmetry of \(R\) and the vertical.

8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .

1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
2. Determine the natural length of the elastic string.

8 A rectangular lamina of mass \(M\) has vertices at the origin \(O ( 0,0 ) , A ( 24 a , 0 ) , B ( 24 a , 6 a )\) and \(C ( 0,6 a )\), where \(a\) is a positive constant. A small object \(P\) of mass \(m\) is attached to the lamina at the point ( \(x , y\) ). The centre of mass of the system consisting of the lamina and \(P\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \(P\) is modelled as a particle and the lamina is modelled as being uniform.

1. Show that \(x = \frac { 12 M a + m x } { M + m }\).
2. Find a corresponding expression for \(\bar { y }\). The lamina, with \(P\) no longer attached, is placed on a horizontal rectangular table, with its sides parallel to the edges of the table, and partly overhanging the edges of the table, as shown in the diagram. The corner of the table is at the point ( \(6 a , 2 a\) ). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-6_538_1431_849_246} When \(P\) is placed on the lamina at \(O\), the lamina topples over one of the edges of the table.
3. Show that \(\mathrm { m } > \frac { 1 } { 2 } \mathrm { M }\). The lamina is now put back on the table in the same position as before. \(P\) is placed at the point \(( 12 a , 6 a )\) on the smooth upper surface of the lamina, and is projected towards \(O\). At a subsequent instant during the motion, \(P\) is at the point (12ak, 6ak) where \(0 < k < 1\).
4. Assuming that the lamina has not yet toppled, find, in terms of \(M\) and \(m\), the value of \(k\) for which the centre of mass of the system lies on the table edge parallel to \(O C\).
5. For the case \(\mathrm { m } = \frac { 3 } { 2 } \mathrm { M }\), determine which table edge the lamina topples over.

4. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of mass 50 kg and length 5 m which overhangs a river by 2 m . When a boy of mass 20 kg stands at \(A\), his sister can walk to within 0.3 m of \(B\), at which point the plank is in limiting equilibrium.

1. What is the mass of the girl?
2. Find the smallest extra weight which must be placed at \(A\) to enable the girl to walk right to the end \(B\).
3. How have you used the fact that the plank is uniform?

1. \begin{figure}[h] \end{figure} Figure 1 shows a non-uniform beam \(A B\) of mass 10 kg and length 6 m resting in a horizontal position on a single support 2 m from \(A\). The beam is supported at \(B\) by a vertical string. Given that the magnitude of the tension in the string is 1.5 times the magnitude of the reaction at the support, find the distance of the centre of mass of the beam from \(A\).
(6 marks)

6. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\) and rolled along the plank to \(B\) without either string breaking.

1. Explain, with the aid of a sketch-graph, how the tension in the string at \(A\) varies with \(x\), the distance of the rock from \(A\).
2. Show that \(M \leq 15\). The first rock is removed and a second rock of mass 20 kg is placed on the plank.
3. Find the fraction of the plank on which the rock can be placed without one of the strings breaking.

4 The diagram shows a uniform lamina \(A B C D E F G H\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-3_346_933_1123_577}

1. Explain why the centre of mass is 25 cm from \(A H\).
2. Show that the centre of mass is 4.375 cm from \(H G\).
3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
4. Explain, briefly, how you have used the fact that the lamina is uniform.

4 A uniform rectangular lamina \(A B C D\) has a mass of 5 kg . The side \(A B\) has length 60 cm and the side \(B C\) has length 30 cm . The points \(P , Q , R\) and \(S\) are the mid-points of the sides, as shown in the diagram below. A uniform triangular lamina \(S R D\), of mass 4 kg , is fixed to the rectangular lamina to form a shop sign. The centre of mass of the triangular lamina \(S R D\) is 10 cm from the side \(A D\) and 5 cm from the side \(D C\). \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-08_613_1086_660_518}

1. Find the distance of the centre of mass of the shop sign from \(A D\).
2. Find the distance of the centre of mass of the shop sign from \(A B\).
3. The shop sign is freely suspended from \(P\). Find the angle between \(A B\) and the horizontal when the shop sign is in equilibrium.
4. To ensure that the side \(A B\) is horizontal when the shop sign is freely suspended from point \(P\), a particle of mass \(m \mathrm {~kg}\) is attached to the shop sign at point \(B\). Calculate \(m\).
5. Explain how you have used the fact that the rectangular lamina \(A B C D\) is uniform in your solution to this question.
   (1 mark)
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-10_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-11_2486_1714_221_153}

8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).

1. Calculate the elastic potential energy when the particle is at the point \(A\).
2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
3. Show that the particle does not remain at rest at the point \(B\).
4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).

4 The diagram shows a uniform lamina which is in the shape of two identical rectangles \(A X G H\) and \(Y B C D\) and a square \(X Y E F\), arranged as shown. The length of \(A X\) is 10 cm , the length of \(X Y\) is 10 cm and the length of \(A H\) is 30 cm . \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-3_1183_1278_513_374}

1. Explain why the centre of mass of the lamina is 15 cm from \(A H\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from the point \(H\). Find, to the nearest degree, the angle between \(H G\) and the horizontal when the lamina is in equilibrium.

9 A smooth hollow hemisphere, of radius \(a\) and centre \(O\), is fixed so that its rim is in a horizontal plane. A smooth uniform \(\operatorname { rod } A B\), of mass \(m\), is in equilibrium, with one end \(A\) resting on the inside of the hemisphere and the point \(C\) on the rod being in contact with the rim of the hemisphere. The rod, of length \(l\), is inclined at an angle \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-6_453_828_559_591}

1. Explain why the reaction between the rod and the hemisphere at point \(A\) acts through \(O\).
2. Draw a diagram to show the forces acting on the rod.
3. Show that \(l = \frac { 4 a \cos 2 \theta } { \cos \theta }\).

3 The diagram shows a uniform rod, \(A B\), of mass 10 kg and length 5 metres. The rod is held in equilibrium in a horizontal position, by a support at \(C\) and a light vertical rope attached to \(A\), where \(A C\) is 2 metres. \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_237_680_479_648}

1. Draw and label a diagram to show the forces acting on the rod.
2. Show that the tension in the rope is 24.5 N .
3. A package of mass \(m \mathrm {~kg}\) is suspended from \(B\). The tension in the rope has to be doubled to maintain equilibrium.
   1. Find \(m\).
   2. Find the magnitude of the force exerted on the rod by the support.
4. Explain how you have used the fact that the rod is uniform in your solution.

2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}

1. Explain why the centre of mass of the lamina lies on \(P Q\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.

4 A uniform plank is 10 m long and has mass 15 kg . It is placed on horizontal ground at the edge of a vertical river bank, so that 2 m of the plank is projecting over the edge, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-3_250_1285_1361_388}

1. A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
2. The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
3. State how you have used the fact that the plank is uniform in your solution.
4. State one other modelling assumption which you have made.

4 Ken is trying to cross a river of width 4 m . He has a uniform plank, \(A B\), of length 8 m and mass 17 kg . The ground on both edges of the river bank is horizontal. The plank rests at two points, \(C\) and \(D\), on fixed supports which are on opposite sides of the river. The plank is at right angles to both river banks and is horizontal. The distance \(A C\) is 1 m , and the point \(C\) is at a horizontal distance of 0.6 m from the river bank. Ken, who has mass 65 kg , stands on the plank directly above the middle of the river, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-3_468_1086_1710_479}

1. Draw a diagram to show the forces acting on the plank.
2. Given that the reaction on the plank at the point \(D\) is \(44 g \mathrm {~N}\), find the horizontal distance of the point \(D\) from the nearest river bank.
3. State how you have used the fact that the plank is uniform in your solution.

3 A uniform rectangular lamina \(A B C D\), of mass 1.6 kg , has side \(A B\) of length 12 cm and side \(B C\) of length 8 cm . To create a logo, a uniform circular lamina, of mass 0.4 kg , is attached. The centre of the circular lamina is at the point \(C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-3_520_780_593_630}

1. Find the distance of the centre of mass of the logo:
   1. from the line \(A B\);
   2. from the line \(A D\).
2. The logo is suspended in equilibrium, with \(A B\) horizontal, by two vertical strings. One string is attached at the point \(A\) and the other string is attached at the point \(B\). Find the tension in each of the two strings.

4 A uniform plank \(A B\), of length 6 m , has mass 25 kg . It is supported in equilibrium in a horizontal position by two vertical inextensible ropes. One of the ropes is attached to the plank at the point \(P\) and the other rope is attached to the plank at the point \(Q\), where \(A P = 1 \mathrm {~m}\) and \(Q B = 0.8 \mathrm {~m}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-2_227_1187_2252_424}

1. 1. Find the tension in each rope.
   2. State how you have used the fact that the plank is uniform in your solution. (1 mark)
2. A particle of mass \(m \mathrm {~kg}\) is attached to the plank at point \(B\), and the tension in each rope is now the same. Find \(m\).

9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks] \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}

5. A uniform rod \(X Y\), of length \(2 a\) and mass \(m\), is connected to a vertical wall by a smooth hinge at the end \(X\). A horizontal light inelastic string connects the mid-point of \(X Y\) to the wall and the rod is in equilibrium in this position.

1. Draw a diagram to show all the forces acting on the rod. Given that the tension in the horizontal string is of magnitude \(2 m g\),
2. find the angle which \(X Y\) makes with the vertical.

8 \begin{figure}[h] \end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height \(h \mathrm {~m}\). The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point \(O\) (see Fig. 1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from \(O\).

1. Show that \(h = 1.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_620_1065_1297_541} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). The tension in the string is \(T \mathrm {~N}\) and the frictional force acting on the object is \(F \mathrm {~N}\).
2. By taking moments about \(O\), express \(F\) in terms of \(T\).
3. Find another equation connecting \(T\) and \(F\). Hence calculate the tension and the frictional force.

8

1. Fig. 1 A uniform lamina \(A B C D\) is in the form of a right-angled trapezium. \(A B = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(A D = 17 \mathrm {~cm}\) (see Fig. 1). Taking \(x\) - and \(y\)-axes along \(A D\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-5_481_1079_991_575} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The lamina is smoothly pivoted at \(A\) and it rests in a vertical plane in equilibrium against a fixed smooth block of height 7 cm . The mass of the lamina is 3 kg . \(A D\) makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the magnitude of the force which the block exerts on the lamina.

3 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-2_497_951_1123_598} A uniform beam \(A B\) has weight 70 N and length 2.8 m . The beam is freely hinged to a wall at \(A\) and is supported in a horizontal position by a strut \(C D\) of length 1.3 m . One end of the strut is attached to the beam at \(C , 0.5 \mathrm {~m}\) from \(A\), and the other end is attached to the wall at \(D\), vertically below \(A\). The strut exerts a force on the beam in the direction \(D C\). The beam carries a load of weight 50 N at its end \(B\) (see diagram).

1. Calculate the magnitude of the force exerted by the strut on the beam.
2. Calculate the magnitude of the force acting on the beam at \(A\).