6.04e Rigid body equilibrium: coplanar forces

1 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_182_843_264_651} A uniform rigid plank has mass 10 kg and length 4 m . The plank has 0.9 m of its length in contact with a horizontal platform. A man \(M\) of mass 75 kg stands on the end of the plank which is in contact with the platform. A child \(C\) of mass 25 kg walks on to the overhanging part of the plank (see diagram). Find the distance between the man and the child when the plank is on the point of tilting.

4 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_737_700_264_721} A uniform beam has length 2.4 m and weight 68 N . The beam is hinged at a fixed point of a vertical wall, and held in a horizontal position by a light rod of length 2.5 m . One end of the rod is attached to the beam at a point 0.7 m from the wall, and the other end of the rod is attached to the wall at a point vertically below the hinge. The beam carries a load of 750 N at its end (see diagram).

1. Find the force in the rod. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally towards the wall and \(Y \mathrm {~N}\) vertically downwards.
2. Find the values of \(X\) and \(Y\).

3 \(A B C D E F\) is the L -shaped cross-section of a uniform solid. This cross-section passes through the centre of mass of the solid and has dimensions as shown in Fig. 1.

1. Find the distance of the centre of mass of the solid from the edge \(A B\) of the cross-section.
   [diagram]
   The solid rests in equilibrium with the face containing the edge \(A F\) of the cross-section in contact with a horizontal table. The weight of the solid is \(W\) N. A horizontal force of magnitude \(P\) N is applied to the solid at the point \(B\), in the direction of \(B C\) (see Fig. 2). The table is sufficiently rough to prevent sliding.
2. Find \(P\) in terms of \(W\), given that the equilibrium of the solid is about to be broken.

6 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-4_620_899_644_623} A rigid rod consists of two parts. The part \(B C\) is in the form of an arc of a circle of radius 2 m and centre \(O\), with angle \(B O C = \frac { 1 } { 4 } \pi\) radians. \(B C\) is uniform and has weight 3 N . The part \(A B\) is straight and of length 2 m ; it is uniform and has weight 4 N . The part \(A B\) of the rod is a tangent to the arc \(B C\) at \(B\). The end \(A\) of the rod is freely hinged to a fixed point of a vertical wall. The rod is held in equilibrium, with the straight part \(A B\) making an angle of \(\frac { 1 } { 4 } \pi\) radians with the wall, by means of a horizontal string attached to \(C\). The string is in the same vertical plane as the rod, and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Show that the centre of mass \(G\) of the part \(B C\) of the rod is at a distance of 2.083 m from the wall, correct to 4 significant figures.
2. Find the value of \(T\).
3. State the magnitude of the horizontal component and the magnitude of the vertical component of the force exerted on the rod by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-5_579_1118_264_516} A particle \(A\) is released from rest at time \(t = 0\), at a point \(P\) which is 7 m above horizontal ground. At the same instant as \(A\) is released, a particle \(B\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(P\) is 24 m . Particle \(B\) moves in the vertical plane containing \(O\) and \(P\), with initial speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle of \(\theta\) above the horizontal (see diagram). Write down
4. an expression for the height of \(A\) above the ground at time \(t \mathrm {~s}\),
5. an expression in terms of \(V , \theta\) and \(t\) for
   1. the horizontal distance of \(B\) from \(O\),
   2. the height of \(B\) above the ground. At time \(t = T\) the particles \(A\) and \(B\) collide at a point above the ground.
   3. Show that \(\tan \theta = \frac { 7 } { 24 }\) and that \(V T = 25\).
   4. Deduce that \(7 V ^ { 2 } > 3125\).

2 A uniform solid cone has height 38 cm .

1. Write down the distance of the centre of mass of the cone from its base. \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840} 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 \(31 ^ { \circ }\) (see diagram), when the cone topples.
2. Find the radius of the cone.
3. Show that \(\mu \geqslant 0.601\), correct to 3 significant figures, where \(\mu\) is the coefficient of friction between the cone and the plane.

5 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_531_791_1633_678} A uniform lamina of weight 15 N has dimensions as shown in the diagram.

1. Show that the distance of the centre of mass of the lamina from \(A B\) is 0.22 m . The lamina is freely hinged at \(B\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(C\). The string passes over a fixed smooth pulley and a particle of mass 1.1 kg is attached to the other end of the string. The lamina is in equilibrium with \(B C\) horizontal. The string is taut and makes an angle of \(\theta ^ { \circ }\) with the horizontal at \(C\), and the particle hangs freely below the pulley (see diagram).
2. Find the value of \(\theta\).

4 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-3_777_447_267_849} A uniform triangular lamina \(A B C\) is right-angled at \(B\) and has sides \(A B = 0.6 \mathrm {~m}\) and \(B C = 0.8 \mathrm {~m}\). The mass of the lamina is 4 kg . One end of a light inextensible rope is attached to the lamina at \(C\). The other end of the rope is attached to a fixed point \(D\) on a vertical wall. The lamina is in equilibrium with \(A\) in contact with the wall at a point vertically below \(D\). The lamina is in a vertical plane perpendicular to the wall, and \(A B\) is horizontal. The rope is taut and at right angles to \(A C\) (see diagram). Find

1. the tension in the rope,
2. the horizontal and vertical components of the force exerted at \(A\) on the lamina by the wall.

2 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_686_495_1238_826} A uniform rigid wire \(A B\) is in the form of a circular arc of radius 1.5 m with centre \(O\). The angle \(A O B\) is a right angle. The wire is in equilibrium, freely suspended from the end \(A\). The chord \(A B\) makes an angle of \(\theta ^ { \circ }\) with the vertical (see diagram).

1. Show that the distance of the centre of mass of the arc from \(O\) is 1.35 m , correct to 3 significant figures.
2. Find the value of \(\theta\).

4 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_375_627_1448_758} Uniform rods \(A B , A C\) and \(B C\) have lengths \(3 \mathrm {~m} , 4 \mathrm {~m}\) and 5 m respectively, and weights \(15 \mathrm {~N} , 20 \mathrm {~N}\) and 25 N respectively. The rods are rigidly joined to form a right-angled triangular frame \(A B C\). The frame is hinged at \(B\) to a fixed point and is held in equilibrium, with \(A C\) horizontal, by means of an inextensible string attached at \(C\). The string is at right angles to \(B C\) and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Find the value of \(T\). A uniform triangular lamina \(P Q R\), of weight 60 N , has the same size and shape as the frame \(A B C\). The lamina is now attached to the frame with \(P , Q\) and \(R\) at \(A , B\) and \(C\) respectively. The composite body is held in equilibrium with \(A , B\) and \(C\) in the same positions as before. Find
2. the new value of \(T\),
3. the magnitude of the vertical component of the force acting on the composite body at \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-2_839_330_539_906} \(A B\) is a diameter of a uniform solid hemisphere with centre \(O\), radius 10 cm and weight 12 N . One end of a light inextensible string is attached to the hemisphere at \(B\) and the other end is attached to a fixed point \(C\) of a vertical wall. The hemisphere is in equilibrium with \(A\) in contact with the wall at a point vertically below \(C\). The centre of mass \(G\) of the hemisphere is at the same horizontal level as \(A\), and angle \(A B C\) is a right angle (see diagram). Calculate the tension in the string.

2
A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). 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 \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.

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

4 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_727_565_1256_790} A uniform lamina of weight 15 N is in the form of a trapezium \(A B C D\) 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 \(A B\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30 ^ { \circ }\) upwards from the horizontal (see diagram). Find the tension in the string.

2 \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-2_319_908_438_616} A uniform solid cone has height 20 cm and base radius \(4 \mathrm {~cm} . P Q\) is a diameter of the base of the cone. The cone is held in equilibrium, with \(P\) in contact with a horizontal surface and \(P Q\) vertical, by a force applied at \(Q\). This force has magnitude 3 N and acts parallel to the axis of the cone (see diagram). Calculate the mass of the cone.

4 \(A B\) is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg . The lamina is hinged to a vertical wall at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. One end of a light inextensible string is attached to the lamina at \(B\) and the other end of the string is attached to the wall vertically above \(A\). The lamina is in equilibrium in a vertical plane perpendicular to the wall with the string horizontal (see diagram).

1. Show that the tension in the string is 2.00 N correct to 3 significant figures.
2. Find the magnitude and direction of the force exerted on the lamina by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-3_956_540_258_804} A small ball \(B\) of mass 0.4 kg is attached to fixed points \(P\) and \(Q\) on a vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(30 ^ { \circ }\) to the vertical. The ball moves in a horizontal circle (see diagram).

2 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829} \(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).

1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between the vertical and the side \(A O\) of the lamina.

5 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-3_348_1205_251_470} \(A B C\) is a uniform triangular lamina of weight 19 N , with \(A B = 0.22 \mathrm {~m}\) and \(A C = B C = 0.61 \mathrm {~m}\). The plane of the lamina is vertical. \(A\) rests on a rough horizontal surface, and \(A B\) is vertical. The equilibrium of the lamina is maintained by a light elastic string of natural length 0.7 m which passes over a small smooth peg \(P\) and is attached to \(B\) and \(C\). The portion of the string attached to \(B\) is horizontal, and the portion of the string attached to \(C\) is vertical (see diagram).

1. Show that the tension in the string is 10 N .
2. Calculate the modulus of elasticity of the string.
3. Find the magnitude and direction of the force exerted by the surface on the lamina at \(A\).

1 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_518_609_255_769} A uniform \(\operatorname { rod } A B\) of weight 16 N is freely hinged at \(A\) to a fixed point. A force of magnitude 4 N acting perpendicular to the rod is applied at \(B\) (see diagram). Given that the rod is in equilibrium,

1. calculate the angle the rod makes with the horizontal,
2. find the magnitude and direction of the force exerted on the rod at \(A\).

2 A uniform lamina \(A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and diameter 0.4 m , and an isosceles triangle \(A B D\) with base \(B D = 0.4 \mathrm {~m}\) and perpendicular height \(h \mathrm {~m}\). The centre of mass of the lamina is at \(O\).

1. Find the value of \(h\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_680_627_1466_797} The lamina is suspended from a vertical string attached to a point \(X\) on the side \(A D\) of the triangle (see diagram). Given the lamina is in equilibrium with \(A D\) horizontal, calculate \(X D\).

3 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_786_1249_1455_447} A smooth hemispherical shell, with centre \(O\), weight 12 N and radius 0.4 m , rests on a horizontal plane. A particle of weight \(W \mathrm {~N}\) lies at rest on the inner surface of the hemisphere vertically below \(O\). A force of magnitude \(F \mathrm {~N}\) acting vertically upwards is applied to the highest point of the hemisphere, which is in equilibrium with its axis of symmetry inclined at \(20 ^ { \circ }\) to the horizontal (see diagram).

1. Show, by taking moments about \(O\), that \(F = 16.48\) correct to 4 significant figures.
2. Find the normal contact force exerted by the plane on the hemisphere in terms of \(W\). Hence find the least possible value of \(W\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-4_597_1011_251_566} \(A B C D E\) is the cross-section through the centre of mass of a uniform prism resting in equilibrium with \(D E\) on a horizontal surface. The cross-section has the shape of a square \(O B C D\) with sides of length \(a \mathrm {~m}\), from which a quadrant \(O A E\) of a circle of radius 1 m has been removed (see diagram).

1. Find the distance of the centre of mass of the prism from \(O\), giving the answer in terms of \(a , \pi\) and \(\sqrt { } 2\).
2. Hence show that $$3 a ^ { 2 } ( 2 - a ) < \frac { 3 } { 2 } \pi - 2$$ and verify that this inequality is satisfied by \(a = 1.68\) but not by \(a = 1.67\).

2 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_481_412_440_865} The diagram shows a circular object formed from a uniform semicircular lamina of weight 11 N and a uniform semicircular arc of weight 9 N . The lamina and the arc both have centre \(O\) and radius 0.7 m and are joined at the ends of their common diameter \(A B\).

1. Show that the distance of the centre of mass of the object from \(O\) is 0.0371 m , correct to 3 significant figures. The object hangs in equilibrium, freely suspended at \(A\).
2. Find the angle between \(A B\) and the vertical and state whether the lowest point of the object is on the lamina or on the arc.

6 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-3_720_723_1165_712} The diagram shows the cross-section \(O A B C D E\) through the centre of mass of a uniform prism. The interior angles of the cross-section at \(O , A , C , D\) and \(E\) are all right angles. \(O A = 0.4 \mathrm {~m} , A B = 0.5 \mathrm {~m}\) and \(B C = C D = 1 \mathrm {~m}\).

1. Calculate the distance of the centre of mass of the prism from \(O E\). The weight of the prism is 120 N . A force of magnitude \(F \mathrm {~N}\) acting along \(D E\) holds the prism in equilibrium when \(O A\) rests on a rough horizontal surface.
2. Find the set of possible values of \(F\).

2 A uniform hemispherical shell of weight 8 N and a uniform solid hemisphere of weight 12 N are joined along their circumferences to form a non-uniform sphere of radius 0.2 m .

1. Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m . This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude \(F \mathrm {~N}\) acting parallel to the axis of symmetry applied to the highest point of the sphere.
2. Calculate \(F\).

6 \includegraphics[max width=\textwidth, alt={}, center]{09971be0-73b6-4c73-8dfd-c89ff877950a-3_451_775_255_685} The diagram shows a uniform lamina \(A B C D E F\), formed from a semicircle with centre \(O\) and radius 1 m by removing a semicircular part with centre \(O\) and radius \(r \mathrm {~m}\).

1. Show that the distance in metres of the centre of mass of the lamina from \(O\) is $$\frac { 4 \left( 1 + r + r ^ { 2 } \right) } { 3 \pi ( 1 + r ) } .$$ The centre of mass of the lamina lies on the \(\operatorname { arc } A B C\).
2. Show that \(r = 0.494\), correct to 3 significant figures. The lamina is freely suspended at \(F\) and hangs in equilibrium.
3. Find the angle between the diameter of the lamina and the vertical.

2 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_448_547_434_799} The diagram shows a uniform object \(A B C\) of weight 3 N in the form of an arc of a circle with centre \(O\) and radius 0.7 m . The angle \(A O C\) is 2 radians. The object rests in equilibrium with \(A\) on a horizontal surface and \(C\) vertically above \(A\). Equilibrium is maintained by a horizontal force of magnitude \(F \mathrm {~N}\) applied at \(C\) in the plane of the object. Calculate \(F\).