Prove MI by integration

4 The region between the curve \(y = \frac { x ^ { 2 } } { a }\) and the \(x\)-axis for \(0 \leqslant x \leqslant a\) is occupied by a uniform lamina with mass \(m\). Show that the moment of inertia of this lamina about the \(x\)-axis is \(\frac { 1 } { 7 } m a ^ { 2 }\).

7 A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\).

1. Prove by integration that the moment of inertia of the rod about an axis through \(P\) perpendicular to \(A B\) is \(\frac { 4 } { 9 } m a ^ { 2 }\). The axis through \(P\) is fixed and horizontal, and the rod can rotate without resistance in a vertical plane about this axis. The rod is released from rest in a horizontal position. Find, in terms of \(m\) and \(g\),
2. the force acting on the rod at \(P\) immediately after the release of the rod,
3. the force acting on the rod at \(P\) at an instant in the subsequent motion when \(B\) is vertically below \(P\).

3 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), is occupied by a uniform lamina of mass 24 kg . The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-2_623_601_1409_706} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2 m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5 m g\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(A B\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(A B\) and the ring \(R\) are in a vertical plane, and \(R C\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac { 1 } { 4 } \pi\), so that the acute angle between \(A B\) and the horizontal is \(2 \theta\) (see diagram).

1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium.
2. Determine whether this position of equilibrium is stable or unstable.

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 \(g\).

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).
3. 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.
4. Show that the lamina begins to rotate anticlockwise.
5. 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).
6. Taking \(O\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a ( \sqrt { 3 } + \sqrt { 3 } \cos 2 \theta + \sin 2 \theta ) .$$
7. Find the two values of \(\theta\) for which the system is in equilibrium.
8. 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).
9. Show that \(\omega ^ { 2 } = 4.8 - 12 \cos \theta\).
10. Find the angular acceleration of the rod in terms of \(\theta\).
11. Show that \(F = 94.8 \cos \theta - 14.4\), and find \(R\) in terms of \(\theta\).
12. 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.

5 A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The point \(C\) on the rod is such that \(A C = a\). The rod can rotate freely in a vertical plane about a fixed horizontal axis passing through \(C\) and perpendicular to the rod.

1. Show by integration that the moment of inertia of the rod about this axis is \(7 m a ^ { 2 }\). The rod starts at rest with \(B\) vertically below \(C\). A couple of constant moment \(\frac { 6 m g a } { \pi }\) is then applied to the rod.
2. Find, in terms of \(a\) and \(g\), the angular speed of the rod when it has turned through one and a half revolutions. \includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-3_721_621_872_762} A light pulley of radius \(a\) is free to rotate in a vertical plane about a fixed horizontal axis passing through its centre \(O\). Two particles, \(P\) of mass \(5 m\) and \(Q\) of mass \(3 m\), are connected by a light inextensible string. The particle \(P\) is attached to the circumference of the pulley, the string passes over the top of the pulley, and \(Q\) hangs below the pulley on the opposite side to \(P\). The section of string not in contact with the pulley is vertical. The fixed line \(O X\) makes an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(O P\) makes an angle \(\theta\) with \(O X\) (see diagram). You are given that the total potential energy of the system (using a suitable reference level) is \(V\), where $$V = m g a ( 3 \sin \theta - 4 \cos \theta - 3 \theta ) .$$

5 The region inside the circle \(x ^ { 2 } + y ^ { 2 } = a ^ { 2 }\) is rotated about the \(x\)-axis to form a uniform solid sphere of radius \(a\) and volume \(\frac { 4 } { 3 } \pi a ^ { 3 }\). The mass of the sphere is \(10 M\).

1. Show by integration that the moment of inertia of the sphere about the \(x\)-axis is \(4 M a ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis.) The sphere is free to rotate about a fixed horizontal axis which is a diameter of the sphere. A particle of mass \(M\) is attached to the lowest point of the sphere. The sphere with the particle attached then makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these oscillations.

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.

2 A uniform solid circular cone has mass \(M\) and base radius \(R\).

1. Show by integration that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M R ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis and that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
2. the constant angular deceleration of the cone,
3. the time taken for the cone to come to rest from the instant that the couple is applied.

4

1. Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter. A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
2. Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.

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

4

1. Show by integration that the moment of inertia of a uniform circular lamina of radius \(a\) and mass \(m\) about an axis perpendicular to the plane of the lamina and through its centre is \(\frac { 1 } { 2 } m a ^ { 2 }\). A closed hollow cylinder has its curved surface and both ends made from the same uniform material. It has mass \(M\), radius \(a\) and height \(h\).
2. Show that the moment of inertia of the cylinder about its axis of symmetry is \(\frac { 1 } { 2 } M a ^ { 2 } \left( \frac { a + 2 h } { a + h } \right)\). For the rest of this question take the cylinder to have mass 8 kg , radius 0.5 m and height 0.3 m .
   The cylinder is at rest and can rotate freely about its axis of symmetry. It is given a tangential impulse of magnitude 55 Ns at a point on its curved surface. The impulse is perpendicular to the axis.
3. Find the angular speed of the cylinder after the impulse. A resistive couple is now applied to the cylinder for 5 seconds. The magnitude of the couple is \(2 \dot { \theta } ^ { 2 } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the cylinder in rad s \({ } ^ { - 1 }\).
4. Formulate a differential equation for \(\dot { \theta }\) and hence find the angular speed of the cylinder at the end of the 5 seconds. The cylinder is now brought to rest by a constant couple of magnitude 0.03 Nm .
5. Calculate the time it takes from when this couple is applied for the cylinder to come to rest.

4 A uniform lamina of mass \(m\) is in the shape of a sector of a circle of radius \(a\) and angle \(\frac { 1 } { 3 } \pi\). It can rotate freely in a vertical plane about a horizontal axis perpendicular to the lamina through its vertex O .

1. Show by integration that the moment of inertia of the lamina about the axis is \(\frac { 1 } { 2 } m a ^ { 2 }\).
2. State the distance of the centre of mass of the lamina from the axis. The lamina is released from rest when one of the straight edges is horizontal as shown in Fig. 4.1. After time \(t\), the line of symmetry of the lamina makes an angle \(\theta\) with the downward vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_257_441_1475_322} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_380_732_1635_1014} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
3. Show that \(\dot { \theta } ^ { 2 } = \frac { 4 g } { \pi a } ( 2 \cos \theta + 1 )\).
4. Find the greatest speed attained by any point on the lamina.
5. Find an expression for \(\ddot { \theta }\) in terms of \(\theta , a\) and \(g\). The lamina strikes a fixed peg at A where \(\mathrm { AO } = \frac { 3 } { 4 } a\) and is horizontal, as shown in Fig. 4.2. The collision reverses the direction of motion of the lamina and halves its angular speed.
6. Find the magnitude of the impulse that the peg gives to the lamina.
7. Determine the maximum value of \(\theta\) in the subsequent motion.

4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
(9)

2. A uniform circular disc has radius \(a\) and mass \(m\). Prove, using integration, that the moment of inertia of the disc about an axis through its centre and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
(Total 5 marks)

4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.

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)

1. (a) Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 a\), about an axis perpendicular to the rod through one end is \(\frac { 4 } { 3 } m a ^ { 2 }\).
   (b) Hence, or otherwise, find the moment of inertia of a uniform square lamina, of mass \(M\) and side \(2 a\), about an axis through one corner and perpendicular to the plane of the lamina.
2. A particle of mass 0.5 kg is at rest at the point with position vector ( \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) ) m. The particle is then acted upon by two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \(( 4 \mathbf { i } + 5 \mathbf { j } - 5 \mathbf { k } ) \mathrm { m }\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\), find \(\mathbf { F } _ { 2 }\).
3. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres at time \(t\) seconds. It is given that \(\mathbf { r }\) satisfies the differential equation

$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t }$$ When \(t = 0 , P\) is at the point with position vector \(3 \mathbf { i }\) metres and is moving with velocity \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
(a) Find \(\mathbf { r }\) in terms of \(t\).
(b) Describe the path of \(P\), giving its cartesian equation.

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 \(g\).

7. Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass \(M\) and base radius \(a\), about its axis is \(\frac { 3 } { 10 } M a ^ { 2 }\).
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\left. \frac { 1 } { 2 } m r ^ { 2 } .\right]\)

1. (a) A uniform lamina of mass \(m\) is in the shape of a triangle \(A B C\). The perpendicular distance of \(C\) from the line \(A B\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(A B\) is \(\frac { 1 } { 6 } m h ^ { 2 }\).
   (b) Deduce the radius of gyration of a uniform square lamina of side \(2 a\), about a diagonal.

The points \(X\) and \(Y\) are the mid-points of the sides \(R Q\) and \(R S\) respectively of a square \(P Q R S\) of side \(2 a\). A uniform lamina of mass \(M\) is in the shape of \(P Q X Y S\).
(c) Show that the moment of inertia of this lamina about \(X Y\) is \(\frac { 79 } { 84 } M a ^ { 2 }\).

4. Show, using integration, that the moment of inertia of a uniform solid right circular cone of mass \(M\), height \(h\) and base radius \(a\), about an axis through the vertex, parallel to the base, is $$\frac { 3 M } { 20 } \left( a ^ { 2 } + 4 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform circular disc, of radius \(r\) and mass \(m\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]

5. \begin{figure}[h] \end{figure} A uniform triangular lamina \(A B C\), of mass \(M\), has \(A B = A C\) and \(B C = 2 a\). The mid-point of \(B C\) is \(D\) and \(A D = h\), as shown in Figure 1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), perpendicular to the plane of the lamina, is $$\frac { M } { 6 } \left( a ^ { 2 } + 3 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform rod, of length \(2 l\) and mass \(m\), about an axis through its midpoint and perpendicular to the rod, is \(\frac { 1 } { 3 } m l ^ { 2 }\).]