6.04e Rigid body equilibrium: coplanar forces

3. The position vector \(\mathbf { r }\) of a particle \(P\) at time \(t\) satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 4 \mathbf { i }$$ Given that the position vector of \(P\) at time \(t = 0\) is \(2 \mathbf { j }\), find the position vector of \(P\) at time \(t\).
(Total 6 marks)

7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).

1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
   (Total 15 marks)

3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),

1. find \(\mathbf { R }\),
2. find a vector equation for the line of action of \(\mathbf { R }\).
   (Total 9 marks) \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
   A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.

5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\). Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,

1. find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical. Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
2. find the period of these oscillations.
   (5)
   (Total 12 marks)

3. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P , P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\frac { 3 } { 4 } \omega\). Find \(x\) in terms of \(a\).
(5)

5. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\).
The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\), and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(3 \mathbf { i } - \mathbf { k }\) ) m. The two forces are equivalent to a single force \(\mathbf { F }\) acting at the point with position vector \(( \mathbf { i } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G }\).

1. Find \(\mathbf { F }\).
2. Find the magnitude of \(\mathbf { G }\).
   (8)

8. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis through the point \(X\) on the rod, where \(A X = \frac { 1 } { 2 } a\). A particle of mass \(m\) is attached to the rod at \(B\). At time \(t = 0\), the rod is vertical, with \(B\) above \(A\), and is given an initial angular speed \(\sqrt { \frac { g } { a } }\). When the rod makes an angle \(\theta\) with the upward vertical, the angular speed of the rod is \(\omega\), as shown in Figure 3.

1. By using the principle of the conservation of energy, show that $$\omega ^ { 2 } = \frac { g } { 2 a } ( 5 - 3 \cos \theta )$$
2. Find the angular acceleration of the rod when it makes an angle \(\theta\) with the upward vertical. When \(\theta = \phi\), the resultant force of the axis on the rod is in a direction perpendicular to the rod.
3. Find \(\cos \phi\).

3. A system of forces consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting on a rigid body. \(\mathbf { F } _ { 1 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\). \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 2 } = ( 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).
Given that the system is equivalent to a single force \(\mathbf { R } \mathrm { N }\), acting at the point with position vector \(( 5 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G N m }\), find

1. \(\mathbf { R }\),
2. the magnitude of \(\mathbf { G }\).
   (9)

5. \begin{figure}[h] \end{figure} A pendulum \(P\) is modelled as a uniform rod \(A B\), of length \(9 a\) and mass \(m\), rigidly fixed to a uniform circular disc of radius \(a\) and mass \(2 m\). The end \(B\) of the rod is attached to the centre of the disc, and the rod lies in the plane of the disc, as shown in Figure 1. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through the end \(A\) and is perpendicular to the plane of the disc.

1. Show that the moment of inertia of \(P\) about \(L\) is \(190 m a ^ { 2 }\). The pendulum makes small oscillations about \(L\).
2. By writing down an equation of motion for \(P\), find the approximate period of these small oscillations.

7. A uniform square lamina \(A B C D\), of mass \(2 m\) and side \(3 a \sqrt { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(24 m a ^ { 2 }\). The lamina is at rest with \(C\) vertically above \(A\). At time \(t = 0\) the lamina is slightly displaced. At time \(t\) the lamina has rotated through an angle \(\theta\).

1. Show that $$2 a \left( \frac { d \theta } { d t } \right) ^ { 2 } = g ( 1 - \cos \theta )$$
2. Show that, at time \(t\), the magnitude of the component of the force acting on the lamina at \(A\), in a direction perpendicular to \(A C\), is \(\frac { 1 } { 2 } m g \sin \theta\). When the lamina reaches the position with \(C\) vertically below \(A\), it receives an impulse which acts at \(C\), in the plane of the lamina and in a direction which is perpendicular to the line \(A C\). As a result of this impulse the lamina is brought immediately to rest.
3. Find the magnitude of the impulse.

4. \begin{figure}[h] \end{figure} A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(O A B\). The angle \(O A B\) is \(90 ^ { \circ } , O A = a\) and \(A B = 2 a\), as shown in Figure 1.

1. Prove, using integration, that the moment of inertia of the lamina \(O A B\) about the edge \(O A\) is \(\frac { 2 } { 3 } M a ^ { 2 }\).
   (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2 l\) about an axis through one end and perpendicular to the rod is \(\frac { 4 } { 3 } m l ^ { 2 }\).) The lamina \(O A B\) is free to rotate about a fixed smooth horizontal axis along the edge \(O A\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120 ^ { \circ }\),
2. find an expression for \(J\), in terms of \(M , a\) and \(g\).

A pendulum consists of a uniform rod \(A B\), of length \(4 a\) and mass \(2 m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(P Q R S\), of mass \(4 m\) and side \(a\). The \(\operatorname { rod } A B\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(A B\) and parallel to the edge \(P Q\) of the square.

1. Show that the moment of inertia of the pendulum about \(L\) is \(75 m a ^ { 2 }\). The pendulum is released from rest when \(B A\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac { 7 } { 24 }\). When \(B A\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(A B\), of the force exerted by the axis \(L\) on the pendulum is \(X\).
2. Find an expression for \(X\) in terms of \(m , g\) and \(\theta\). Using the approximation \(\theta \approx \sin \theta\),
3. find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position.

2. \begin{figure}[h] \end{figure} A uniform circular disc has mass \(4 m\), centre \(O\) and radius \(4 a\). The line \(P O Q\) is a diameter of the disc. A circular hole of radius \(2 a\) is made in the disc with the centre of the hole at the point \(R\) on \(P Q\) where \(Q R = 5 a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.

1. Show that the moment of inertia of the lamina about \(L\) is \(69 m a ^ { 2 }\). The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac { 2 \pi } { 3 }\) before it first comes to instantaneous rest,
2. find \(\Omega\) in terms of \(g\) and \(a\).

A uniform lamina \(A B C\) of mass \(m\) is in the shape of an isosceles triangle with \(A B = A C = 5 a\) and \(B C = 8 a\).

1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(B C\), is \(\frac { 9 } { 2 } m a ^ { 2 }\). The foot of the perpendicular from \(A\) to \(B C\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(D A\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(A\), perpendicular to \(B C\) and in the plane of the lamina, is \(\frac { 8 } { 3 } m a ^ { 2 }\).
2. Find the angular acceleration of the lamina when \(D A\) makes an angle \(\theta\) with the downward vertical. Given that \(\alpha\) is small,
3. find an approximate value for the period of oscillation of the lamina about the vertical.

A uniform circular disc has mass \(m\), centre \(O\) and radius \(2 a\). It is free to rotate about a fixed smooth horizontal axis \(L\) which lies in the same plane as the disc and which is tangential to the disc at the point \(A\). The disc is hanging at rest in equilibrium with \(O\) vertically below \(A\) when it is struck at \(O\) by a particle of mass \(m\). Immediately before the impact the particle is moving perpendicular to the plane of the disc with speed \(3 \sqrt { } ( a g )\). The particle adheres to the disc at \(O\).

1. Find the angular speed of the disc immediately after the impact.
2. Find the magnitude of the force exerted on the disc by the axis immediately after the impact.

6. \begin{figure}[h] \end{figure} A light inextensible string has a particle of mass \(m\) attached to one end and a particle of mass \(4 m\) attached to the other end. The string passes over a rough pulley which is modelled as a uniform circular disc of radius \(a\) and mass \(2 m\), as shown in Figure 2. The pulley can rotate in a vertical plane about a fixed horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. As the pulley rotates, a frictional couple of constant magnitude \(2 m g a\) acts on it. The system is held with the string vertical and taut on each side of the pulley and released from rest. Given that the string does not slip on the pulley, find the initial angular acceleration of the pulley.

6. Three equal uniform rods, each of mass \(m\) and length \(2 a\), form the sides of a rigid equilateral triangular frame \(A B C\). The frame is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the frame.

1. Show that the moment of inertia of the frame about \(L\) is \(6 m a ^ { 2 }\). The frame is held with \(A B\) horizontal and \(C\) below \(A B\), and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,
2. find the magnitude of the force exerted by the axis on the frame at \(A\) at the instant when the frame is released.

4 A uniform lamina ABDE is in the shape of an equilateral triangle ABC of side 12 cm from which an equilateral triangle of side 6 cm has been removed from corner \(C\). The lamina is situated on coordinate axes as shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Explain why angle \(\mathrm { BDA } = 90 ^ { \circ }\).
2. Find the coordinates of the centre of mass of the lamina ABDE . The lamina ABDE is now freely suspended from D and hangs in equilibrium.
3. Calculate the angle DE makes with the downward vertical.

4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle \(\alpha\) with \(\mathrm { O } y\), where \(\sin \alpha = \frac { 7 } { 25 }\). The rod and lamina are rigidly attached at E, the mid-point of CD.
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg . \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Find,
   1. the \(y\)-coordinate of the centre of mass of the shovel,
   2. the \(z\)-coordinate of the centre of mass of the shovel. The shovel is freely suspended from O and hangs in equilibrium.
2. Calculate the angle that OE makes with the vertical.

5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h] \end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
   \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
   \end{figure} The weight of S is \(W \mathrm {~N}\).
3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .

6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h] \end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.

1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
   \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
   \section*{END OF QUESTION PAPER}

3 Fig. 3 shows a light square lamina ABCD , of side length 0.75 m , suspended vertically by wires attached to A and B so that AB is horizontal. A particle P of mass \(m \mathrm {~kg}\) is attached to the edge DC . The lamina hangs in equilibrium. \begin{figure}[h] \end{figure} The tension in the wire attached to A is 14 N and the tension in the wire attached to B is \(T \mathrm {~N}\). The wire at A makes an angle of \(25 ^ { \circ }\) with the horizontal and the wire at B makes an angle of \(60 ^ { \circ }\) with the horizontal.

1. Determine the value of \(T\).
2. Determine
   1. the value of \(m\),
   2. the distance of P from D . P is moved to the midpoint of CD . A couple is applied to the lamina so that it remains in equilibrium with AB horizontal and the tension in both wires unchanged.
3. Determine

4 Fig. 4 shows a uniform beam of length \(2 a\) and weight \(W\) leaning against a block of weight \(2 W\) which is on a rough horizontal plane. The beam is freely hinged to the plane at O and makes an angle \(\theta\) with the horizontal. The contact between the beam and the block is smooth. The beam and block are in equilibrium, and it may be assumed that the block does not topple. \begin{figure}[h] \end{figure} Let

• \(S\) be the normal contact force between the beam and the block,
• \(R\) be the normal contact force between the plane and the block,
• \(F\) be the frictional force between the plane and the block.

Partially complete force diagrams showing the beam and the block separately are given in the Printed Answer Booklet.

1. Add the forces listed above to these diagrams. It is given that \(\theta = 30 ^ { \circ }\).
2. Determine the minimum possible value of the coefficient of friction between the block and the plane.
3. In each case explain, with justification, how your answer to part (b) would change (assuming the rest of the system remained unchanged) if
   1. \(\theta < 30 ^ { \circ }\),
   2. the contact between the beam and the block were rough.

6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).

1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.

6 Fig. 6.1 shows a cross-section through a block of mass 5 kg which is on top of a trolley of mass 11 kg . The trolley is on top of a smooth horizontal surface. The coefficient of friction between the block and the trolley is 0.3 . Throughout this question you may assume that there are no other resistances to motion on either the block or the trolley. \begin{figure}[h] \end{figure} Initially, both the block and trolley are at rest. A constant force of magnitude 50 N is now applied horizontally to the trolley, as shown in Fig. 6.1.

1. Show that in the subsequent motion the block will slide.
2. Find the acceleration of
   1. the block,
   2. the trolley. The same block and trolley are again at rest. An obstruction, in the form of a fixed horizontal pole, is placed in front of the block, the pole is 91 cm above the trolley and the width of the block is 56 cm as shown in Fig. 6.2, as well as the forward direction of motion. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-6_426_1324_1793_269} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
      \end{figure} It is given that the block is uniform and that the contact between the pole and the block is smooth. A small horizontal force is now applied to the trolley in the forward direction of motion and gradually increased.
3. Determine whether the block will topple or slide.