6.04d Integration: for centre of mass of laminas/solids

3 \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-2_392_746_908_645} A non-uniform rectangular lamina \(A B C D\) has mass 6 kg . The centre of mass \(G\) of the lamina is 0.8 m from the side \(A D\) and 0.5 m from the side \(A B\) (see diagram). The moment of inertia of the lamina about \(A D\) is \(6.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\) and the moment of inertia of the lamina about \(A B\) is \(2.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina rotates in a vertical plane about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.

1. Write down the moment of inertia of the lamina about this axis. The lamina is released from rest in the position where \(A B\) and \(D C\) are horizontal and \(D C\) is above \(A B\). A frictional couple of constant moment opposes the motion. When \(A B\) is first vertical, the angular speed of the lamina is \(2.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the moment of the frictional couple.
3. Find the angular acceleration of the lamina immediately after it is released.

4 \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-3_698_505_275_801} A uniform solid cylinder has radius \(a\), height \(3 a\), and mass \(M\). The line \(A B\) is a diameter of one of the end faces of the cylinder (see diagram).

1. Show by integration that the moment of inertia of the cylinder about \(A B\) is \(\frac { 13 } { 4 } M a ^ { 2 }\). (You may assume that the moment of inertia of a uniform disc of mass \(m\) and radius \(a\) about a diameter is \(\frac { 1 } { 4 } m a ^ { 2 }\).) The line \(A B\) is now fixed in a horizontal position and the cylinder rotates freely about \(A B\), making small oscillations as a compound pendulum.
2. Find the approximate period of these small oscillations, in terms of \(a\) and \(g\).

3 The region bounded by the curve \(y = 2 x + x ^ { 2 }\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, and the line \(x = 3\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

5 The region bounded by the curve \(y = \sqrt { a x }\) for \(a \leqslant x \leqslant 4 a\) (where \(a\) is a positive constant), the \(x\)-axis, and the lines \(x = a\) and \(x = 4 a\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution of mass \(m\).

1. Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 7 } { 5 } m a ^ { 2 }\). The solid is free to rotate about a fixed horizontal axis along the line \(y = a\), and makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these small oscillations. \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-3_734_862_813_644} A uniform rectangular lamina \(A B C D\) has mass \(m\) and sides \(A B = 2 a\) and \(B C = 3 a\). The mid-point of \(A B\) is \(P\) and the mid-point of \(C D\) is \(Q\). The lamina is rotating freely in a vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the point \(X\) on \(P Q\) where \(P X = a\). Air resistance may be neglected. When \(Q\) is vertically above \(X\), the angular speed is \(\sqrt { \frac { 9 g } { 10 a } }\). When \(X Q\) makes an angle \(\theta\) with the upward vertical, the angular speed is \(\omega\), and the force acting on the lamina at \(X\) has components \(R\) parallel to \(P Q\) and \(S\) parallel to \(B A\) (see diagram).

2 The region \(R\) is bounded by the \(x\)-axis, the lines \(x = a\) and \(x = 2 a\), and the curve \(y = \frac { a ^ { 3 } } { x ^ { 2 } }\) for \(a \leqslant x \leqslant 2 a\), where \(a\) is a positive constant. A uniform solid of revolution is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

4 From a helicopter, a small plane is spotted 3750 m away on a bearing of \(075 ^ { \circ }\). The plane is at the same altitude as the helicopter, and is flying with constant speed \(62 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal straight line on a bearing of \(295 ^ { \circ }\). The helicopter flies with constant speed \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line, and intercepts the plane.

1. Find the bearings of the two possible directions in which the helicopter could fly.
2. Given that interception occurs in the shorter of the two possible times, find the time taken to make the interception. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-3_668_298_260_922} A uniform lamina of mass 63 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, and the curve \(y = 8 - x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\). The unit of length is the metre. The vertices of the lamina are \(O ( 0,0 )\), \(A ( 2,0 )\) and \(B ( 0,8 )\) (see diagram).

1. Show that the moment of inertia of this lamina about \(O B\) is \(56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). It is given that the moment of inertia of the lamina about \(O A\) is \(1036.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the centre of mass of the lamina has coordinates \(\left( \frac { 4 } { 5 } , \frac { 24 } { 7 } \right)\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis passing through \(O\) and perpendicular to the lamina. Starting with the lamina at rest with \(B\) vertically above \(O\), a couple of constant anticlockwise moment 800 Nm is applied to the lamina.
2. Show that the lamina begins to rotate anticlockwise.
3. Find the angular speed of the lamina at the instant when \(O B\) first becomes horizontal. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-4_709_752_267_699} A smooth circular wire, with centre \(O\) and radius \(a\), is fixed in a vertical plane, and the point \(A\) is on the wire at the same horizontal level as \(O\). A small bead \(B\) of mass \(m\) can move freely on the wire. A light elastic string, with natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\), passes through a fixed ring at \(A\), and has one end fixed at \(O\) and the other end attached to \(B\). The section \(A B\) of the string is at an angle \(\theta\) above the horizontal, where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), so that \(O B\) is at an angle \(2 \theta\) to the horizontal (see diagram).

1. Taking \(O\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a ( \sqrt { 3 } + \sqrt { 3 } \cos 2 \theta + \sin 2 \theta ) .$$
2. Find the two values of \(\theta\) for which the system is in equilibrium.
3. For each position of equilibrium, determine whether it is stable or unstable. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-5_478_1403_267_372} A thin horizontal rail is fixed at a height of 0.6 m above horizontal ground. A non-uniform straight \(\operatorname { rod } A B\) has mass 6 kg and length 3 m ; its centre of mass \(G\) is 2 m from \(A\) and 1 m from \(B\), and its moment of inertia about a perpendicular axis through its mid-point \(M\) is \(4.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The rod is placed in a vertical plane perpendicular to the rail, with \(A\) on the ground and \(M\) in contact with the rail. It is released from rest in this position, and begins to rotate about \(M\), without slipping on the rail. When the angle between \(A B\) and the upward vertical is \(\theta\) radians, the rod has angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\), the frictional force in the direction \(A B\) is \(F \mathrm {~N}\), and the normal reaction is \(R \mathrm {~N}\) (see diagram).

1. Show that \(\omega ^ { 2 } = 4.8 - 12 \cos \theta\).
2. Find the angular acceleration of the rod in terms of \(\theta\).
3. Show that \(F = 94.8 \cos \theta - 14.4\), and find \(R\) in terms of \(\theta\).
4. Given that the coefficient of friction between the rod and the rail is 0.9 , show that the rod will slip on the rail before \(B\) hits the ground.

2 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 3\), and the curve \(y = \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant \ln 3\), is occupied by a uniform lamina. Find, in an exact form, the coordinates of the centre of mass of this lamina.

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is \(k \mathrm { e } ^ { - \frac { x } { a } }\), where \(k\) is a constant. Find, in an exact form, the distance of the centre of mass of the rod from \(A\).

2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis and the curve \(y = x \left( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } \right)\) for \(0 \leqslant x \leqslant a\), where \(a\) is a constant, about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

4 A uniform lamina of mass 18 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 9\) and the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln 9\). The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.

4 A flagpole AB of length \(2 a\) is modelled as a thin rigid rod of variable mass per unit length given by $$\rho = \frac { M } { 8 a ^ { 2 } } ( 5 a - x ) ,$$ where \(x\) is the distance from A and \(M\) is the mass of the flagpole.

1. Show that the moment of inertia of the flagpole about an axis through A and perpendicular to the flagpole is \(\frac { 7 } { 6 } M a ^ { 2 }\). Show also that the centre of mass of the flagpole is at a distance \(\frac { 11 } { 12 } a\) from A . The flagpole is hinged to a wall at A and can rotate freely in a vertical plane. A light inextensible rope of length \(2 \sqrt { 2 } a\) is attached to the end B and the other end is attached to a point on the wall a distance \(2 a\) vertically above A, as shown in Fig. 4. The flagpole is initially at rest when lying vertically against the wall, and then is displaced slightly so that it falls to a horizontal position, at which point the rope becomes taut and the flagpole comes to rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-4_403_365_1174_849} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
2. Find an expression for the angular velocity of the flagpole when it has turned through an angle \(\theta\).
3. Show that the vertical component of the impulse in the rope when it becomes taut is \(\frac { 1 } { 12 } M \sqrt { 77 a g }\). Hence write down the horizontal component.
4. Find the horizontal and vertical components of the impulse that the hinge exerts on the flagpole when the rope becomes taut. Hence find the angle that this impulse makes with the horizontal.

3

1. Show, by integration, that the moment of inertia of a uniform rod of mass \(m\) and length \(2 a\) about an axis through its centre and perpendicular to the rod is \(\frac { 1 } { 3 } m a ^ { 2 }\). A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod. The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
3. The pendulum is held with AB at an angle \(\alpha\) to the downward vertical and released from rest. At time \(t , \mathrm { AB }\) is at an angle \(\theta\) to the vertical. Find an expression for \(\dot { \theta } ^ { 2 }\) in terms of \(\theta\) and \(\alpha\).
4. Hence, or otherwise, show that, provided that \(\alpha\) is small, the pendulum performs simple harmonic motion. Calculate the period.

3 A circular disc of radius \(a \mathrm {~m}\) has mass per unit area \(\rho \mathrm { kg } \mathrm { m } ^ { - 2 }\) given by \(\rho = k ( a + r )\), where \(r \mathrm {~m}\) is the distance from the centre and \(k\) is a positive constant. The disc can rotate freely about an axis perpendicular to it and through its centre.

1. Show that the mass, \(M \mathrm {~kg}\), of the disc is given by \(M = \frac { 5 } { 3 } k \pi a ^ { 3 }\), and show that the moment of inertia, \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), about this axis is given by \(I = \frac { 27 } { 50 } M a ^ { 2 }\). For the rest of this question, take \(M = 64\) and \(a = 0.625\).
   The disc is at rest when it is given a tangential impulsive blow of 50 N s at a point on its circumference.
2. Find the angular speed of the disc. The disc is then accelerated by a constant couple reaching an angular speed of \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 20 seconds.
3. Calculate the magnitude of this couple. When the angular speed is \(30 \mathrm { rads } ^ { - 1 }\), the couple is removed and brakes are applied to bring the disc to rest. The effect of the brakes is modelled by a resistive couple of \(3 \dot { \theta } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the disc in \(\mathrm { rad } \mathrm { s } ^ { - 1 }\).
4. Formulate a differential equation for \(\dot { \theta }\) and hence find \(\dot { \theta }\) in terms of \(t\), the time in seconds from when the brakes are first applied.
5. By reference to your expression for \(\dot { \theta }\), give a brief criticism of this model for the effect of the brakes.

3 A uniform circular disc has mass \(M\) and radius \(a\). The centre of the disc is at point C .

1. Show by integration that the moment of inertia of the disc about an axis through C and perpendicular to the disc is \(\frac { 1 } { 2 } M a ^ { 2 }\). The point A on the disc is at a distance \(\frac { 1 } { 10 } a\) from its centre.
2. Show that the moment of inertia of the disc about an axis through A and perpendicular to the disc is \(0.51 M a ^ { 2 }\). The disc can rotate freely in a vertical plane about an axis through A that is horizontal and perpendicular to the disc. The disc is held slightly displaced from its stable equilibrium position and is released from rest. In the motion that follows, the angle that AC makes with the downward vertical is \(\theta\).
3. Write down the equation of motion for the disc. Assuming \(\theta\) remains sufficiently small throughout the motion, show that the disc performs approximate simple harmonic motion and determine the period of the motion. A particle of mass \(m\) is attached at a point P on the circumference of the disc, so that the centre of mass of the system is now at A .
4. Sketch the position of P in relation to A and C . Find \(m\) in terms of \(M\) and show that the moment of inertia of the system about the axis through A and perpendicular to the disc is \(0.6 M a ^ { 2 }\). The system now rotates at a constant angular speed \(\omega\) about the axis through A .
5. Find the kinetic energy of the system. Hence find the magnitude of the constant resistive couple needed to bring the system to rest in \(n\) revolutions.

4 In this question you may assume without proof the standard results in Examination Formulae and Tables (MF2) for

• the moment of inertia of a disc about an axis through its centre perpendicular to the disc,
• the position of the centre of mass of a solid uniform cone.

Fig. 4 shows a uniform cone of radius \(a\) and height \(2 a\), with its axis of symmetry on the \(x\)-axis and its vertex at the origin. A thin slice through the cone parallel to the base is at a distance \(x\) from the vertex. \begin{figure}[h] \end{figure} The slice is taken to be a thin uniform disc of mass \(m\).

1. Write down the moment of inertia of the disc about the \(x\)-axis. Hence show that the moment of inertia of the disc about the \(y\)-axis is \(\frac { 17 } { 16 } m x ^ { 2 }\).
2. Hence show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 51 } { 20 } M a ^ { 2 }\), where \(M\) is the mass of the cone. [You may assume without proof the formula for the volume of a cone.] The cone is now suspended so that it can rotate freely about a fixed, horizontal axis through its vertex. The axis of symmetry of the cone moves in a vertical plane perpendicular to the axis of rotation. The cone is released from rest when its axis of symmetry is at an acute angle \(\alpha\) to the downward vertical. At time \(t\), the angle the axis of symmetry makes with the downward vertical is \(\theta\).
3. Use an energy method to show that \(\dot { \theta } ^ { 2 } = \frac { 20 g } { 17 a } ( \cos \theta - \cos \alpha )\).
4. Hence, or otherwise, show that if \(\alpha\) is small the cone performs approximate simple harmonic motion and find the period. RECOGNISING ACHIEVEMENT

7. \begin{figure}[h] \end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).

1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END

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)

7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.

1. Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\). The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
2. Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
3. Find the greatest magnitude of the angular acceleration during the motion.
   (Total 17 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)

4. \begin{figure}[h] \end{figure} A region \(R\) is bounded by the curve \(y ^ { 2 } = 4 a x ( y > 0 )\), the \(x\)-axis and the line \(x = a ( a > 0 )\), as shown in Figure 1. A uniform solid \(S\) of mass \(M\) is formed by rotating \(R\) about the \(x\)-axis through \(360 ^ { \circ }\). Using integration, prove that the moment of inertia of \(S\) about the \(x\)-axis is \(\frac { 4 } { 3 } M a ^ { 2 }\).
(You may assume without proof that the moment of inertia of a uniform disc, of mass \(m\) and radius \(r\), about an axis through its centre perpendicular to its plane is \(\frac { 1 } { 2 } m r ^ { 2 }\).)

6. \begin{figure}[h] \end{figure} A lamina \(S\) is formed from a uniform disc, centre \(O\) and radius \(2 a\), by removing the disc of centre \(O\) and radius \(a\), as shown in Figure 2. The mass of \(S\) is \(M\).

1. Show that the moment of inertia of \(S\) about an axis through \(O\) and perpendicular to its plane is \(\frac { 5 } { 2 } M a ^ { 2 }\).
   (3) The lamina is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) lies in the plane of \(S\) and is a tangent to its outer circumference, as shown in Figure 2.
2. Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 21 } { 4 } M a ^ { 2 }\).
   (4) \(S\) is displaced through a small angle from its position of stable equilibrium and, at time \(t = 0\), it is released from rest. Using the equation of motion of \(S\), with a suitable approximation,
3. find the time when \(S\) first passes through its position of stable equilibrium.
   (6)

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 \(\cos \phi\).

6. A uniform solid right circular cylinder has mass \(M\), height \(h\) and radius \(a\). Find, using integration, its moment of inertia about a diameter of one of its circular ends.
[0pt] [You may assume without proof that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about a diameter is \(\frac { 1 } { 4 } m a ^ { 2 }\).]

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.