3.04b Equilibrium: zero resultant moment and force

5 \includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_533_698_1343_721} A uniform rectangular lamina \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(M\). A uniform circular lamina of radius \(\frac { 5 } { 2 } a\) has mass \(\frac { 1 } { 3 } M\). The two laminas are fixed together in the same plane with their centres coinciding at the point \(O\) (see diagram). A particle \(P\) of mass \(\frac { 1 } { 2 } M\) is attached at \(C\). The system is free to rotate about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane \(A B C D\). Show that the moment of inertia of the system about this axis is \(\frac { 2225 } { 24 } M a ^ { 2 }\). The system is released from rest with \(A C\) horizontal and \(D\) below \(A C\). Find, in the form \(k \sqrt { } \left( \frac { g } { a } \right)\), the greatest angular speed in the subsequent motion, giving the value of \(k\) correct to 3 decimal places.
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2 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_316_1065_712_541} Two identical uniform heavy triangular prisms, each of base width 10 cm , are arranged as shown at the ends of a smooth horizontal shelf of length 1 m . Some books, each of width 5 cm , are placed on the shelf between the prisms.

1. Find how far the base of a prism can project beyond an end of the shelf without the prism toppling.
2. Find the greatest number of books that can be stored on the shelf without either of the prisms toppling.

3 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_202_972_1619_584} A light elastic string has natural length 0.8 m and modulus of elasticity 12 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.96 m apart. A particle of weight \(W \mathrm {~N}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.14 m below \(A B\) (see diagram). Find \(W\).

5 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_590_754_1425_699} A uniform lamina of weight 9 N has dimensions as shown in the diagram. The lamina is freely hinged to a fixed point at \(A\). A light inextensible string has one end attached to \(B\), and the other end attached to a fixed point \(C\), which is in the same vertical plane as the lamina. The lamina is in equilibrium with \(A B\) horizontal and angle \(A B C = 150 ^ { \circ }\).

1. Show that the tension in the string is 12.2 N .
2. Find the magnitude of the force acting on the lamina at \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-3_464_439_274_854} A uniform beam \(A B\) has length 6 m and mass 45 kg . One end of a light inextensible rope is attached to the beam at the point 2.5 m from \(A\). The other end of the rope is attached to a fixed point \(P\) on a vertical wall. The beam is in equilibrium with \(A\) in contact with the wall at a point 5 m below \(P\). The rope is taut and at right angles to \(A B\) (see diagram). Find

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

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 \(7 V ^ { 2 } > 3125\).

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.

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.

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).

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 \(A\).

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]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641} \(O A B C\) is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre \(O\), radius \(O A = r \mathrm {~m}\) and angle \(A O C = \frac { 2 } { 3 } \pi\) radians. The prism lies on a plane inclined at an angle \(\theta\) radians to the horizontal, where \(\theta < \frac { 1 } { 3 } \pi\). OC lies along a line of greatest slope with \(O\) higher than \(C\). The prism is freely hinged to the plane at \(O\). A force of magnitude 15 N acts at \(A\), in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of \(\theta\).

6 \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-3_474_860_1288_644} A uniform solid cone of height 1.2 m and semi-vertical angle \(\theta ^ { \circ }\) is divided into two parts by a cut parallel to and 0.4 m from the circular base. The upper conical part, \(C\), has weight 16 N , and the lower part, \(L\), has weight 38 N . The two parts of the solid rest in equilibrium with the larger plane face of \(L\) on a horizontal surface and the smaller plane face of \(L\) covered by the base of \(C\) (see diagram).

1. Calculate the distance of the centre of mass of \(L\) from its larger plane face. An increasing horizontal force is applied to the vertex of \(C\). Equilibrium is broken when the magnitude of this force first exceeds 4 N , and \(C\) begins to slide on \(L\).
2. By considering the forces on \(C\),
   (a) find the coefficient of friction between \(C\) and \(L\),
   (b) show that \(\theta > 14.0\), correct to 3 significant figures. \(C\) is removed and \(L\) is placed with its curved surface on the horizontal surface.
3. Given that \(L\) is on the point of toppling, calculate \(\theta\).

5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the greatest speed of the particle during its descent.
2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
3. Find the distance of the centre of mass of the rod from \(A\).
   The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
4. Calculate \(P\).
   In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
   When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
7. Calculate \(V\) and \(a\).
8. Find the direction of motion of \(P\) when it passes through \(A\).

6 \(A B C\) is a uniform lamina in the form of a triangle with \(A B = 0.3 \mathrm {~m} , B C = 0.6 \mathrm {~m}\) and a right angle at \(B\) (see diagram).

1. State the distances of the centre of mass of the lamina from \(A B\) and from \(B C\). Distance from \(A B\) Distance from \(B C\) \(\_\_\_\_\) The lamina is freely suspended at \(B\) and hangs in equilibrium.
2. Find the angle between \(A B\) and the horizontal.
   A force of magnitude 12 N is applied along the edge \(A C\) of the lamina in the direction from \(A\) towards \(C\). The lamina, still suspended at \(B\), is now in equilibrium with \(A B\) vertical.
3. Calculate the weight of the lamina.

5 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_383_1031_543_557} A non-uniform rod \(A B\) of length 2.5 m and mass 3 kg has its centre of mass at the point \(G\) of the rod, where \(A G = 1.5 \mathrm {~m}\). The rod hangs horizontally, in equilibrium, from strings attached at \(A\) and \(B\). The strings at \(A\) and \(B\) make angles with the vertical of \(\alpha ^ { \circ }\) and \(15 ^ { \circ }\) respectively. The tension in the string at \(B\) is \(T \mathrm {~N}\) (see diagram). Find

1. the value of \(T\),
2. the value of \(\alpha\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).

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