Moments of inertia

4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end.
(10)

1. A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector ( \(2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) ) m relative to a fixed origin \(O\) to a point \(B\) with position vector ( \(3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k }\) ) m relative to \(O\) under the action of a force \(\mathbf { F } = ( 14 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\) N. Find
   1. the work done by \(\mathbf { F }\) as the bead moves from \(A\) to \(B\),
   2. the speed of the bead at \(B\).
   3. (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 its centre is \(\frac { 1 } { 3 } m a ^ { 2 }\).
      (3)
   A uniform wire of mass \(4 m\) and length \(8 a\) is bent into the shape of a square.
2. Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane.
   (4)

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 \(\frac { 1 } { 4 } m r ^ { 2 }\).]

1. Show that the moment of inertia of the object about the axis \(l\) is \(180 M a ^ { 2 }\).
2. Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.

5 A uniform circular disc has diameter \(A B\), mass \(2 m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac { 13 } { 2 } m a ^ { 2 }\). The disc is held with \(A B\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(A B\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position.

5 A uniform rectangular thin sheet of glass \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(\frac { 3 } { 5 } M\). Each of the edges \(A B , B C , C D\) and \(D A\) has a thin strip of metal attached to it, as a border to the glass. The strips along \(A B\) and \(C D\) each have mass \(M\), and the strips along \(B C\) and \(D A\) each have mass \(\frac { 1 } { 3 } M\). Show that the moment of inertia of the whole object (glass and metal strips) about an axis through \(A\) perpendicular to the plane of the object is \(128 M a ^ { 2 }\). The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with \(C\) vertically below \(A\). It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion.

1 A uniform wire, of length \(24 a\) and mass \(m\), is bent into the form of a triangle \(A B C\) with angle \(A B C = 90 ^ { \circ }\), \(A B = 6 a\) and \(B C = 8 a\) (see diagram). Find the moment of inertia of the wire about an axis through \(A\) perpendicular to the plane of the wire.

2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
   In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
2. Find the value of \(e\).
3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{89a83576-4568-46c8-8872-f59f2397627d-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
4. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
   The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
5. Find the possible values of \(x\).

A uniform disc, with centre \(O\) and radius \(a\), is surrounded by a uniform concentric ring with radius \(3 a\). The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass \(\frac { 3 } { 2 } m\) and length \(2 a\). The disc and the ring each have mass \(2 m\). The rods meet the ring at the points \(A , B , C\) and \(D\). The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through \(O\) perpendicular to the plane of the object is \(45 m a ^ { 2 }\). Find the moment of inertia of the object about an axis \(l\) through \(A\) in the plane of the object and tangential to the ring. A particle of mass \(3 m\) is now attached to the object at \(C\). The object, including the additional particle, is suspended from the point \(A\) and hangs in equilibrium. It is free to rotate about the axis \(l\). The centre of the disc is given a horizontal speed \(u\). When, in the subsequent motion, the object comes to instantaneous rest, \(C\) is below the level of \(A\) and \(A C\) makes an angle \(\sin ^ { - 1 } \left( \frac { 1 } { 4 } \right)\) with the horizontal. Find \(u\) in terms of \(a\) and \(g\).

1. Show that the moment of inertia of the lamina about the axis through \(X\) is \(\frac { 4 } { 3 } m a ^ { 2 }\).
2. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(\omega ^ { 2 } = \frac { 6 g } { 5 a }\).
3. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(R = 0\), and given also that \(\sin \theta = \frac { 4 } { 5 }\) find \(S\) in terms of \(m\) and \(g\).

6 A rigid body consists of a uniform rod \(A B\), of mass 15 kg and length 2.8 m , with a particle of mass 5 kg attached at \(B\). The body rotates without resistance in a vertical plane about a fixed horizontal axis through \(A\).

1. Find the distance of the centre of mass of the body from \(A\).
2. Find the moment of inertia of the body about the axis.
   \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-3_475_682_680_719} At one instant, \(A B\) makes an acute angle \(\theta\) with the downward vertical, the angular speed of the body is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular acceleration of the body is \(3.5 \mathrm { rad } \mathrm { s } ^ { - 2 }\) (see diagram).
3. Show that \(\sin \theta = 0.8\).
4. Find the components, parallel and perpendicular to \(B A\), of the force acting on the body at \(A\).
   [0pt] [Question 7 is printed overleaf.]
   \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-4_949_1112_281_550} A small bead \(B\), of mass \(m\), slides on a smooth circular hoop of radius \(a\) and centre \(O\) which is fixed in a vertical plane. A light elastic string has natural length \(2 a\) and modulus of elasticity \(m g\); one end is attached to \(B\), and the other end is attached to a light ring \(R\) which slides along a smooth horizontal wire. The wire is in the same vertical plane as the hoop, and at a distance \(2 a\) above \(O\). The elastic string \(B R\) is always vertical, and \(O B\) makes an angle \(\theta\) with the downward vertical (see diagram).
5. Show that \(\theta = 0\) is a position of stable equilibrium.
6. Find the approximate period of small oscillations about the equilibrium position \(\theta = 0\).

7
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina \(A B C D\) with \(A B = 6 a , A D = 8 a\) and centre \(G\). The mass of the lamina is \(m\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\) and perpendicular to the lamina.

1. Find the moment of inertia of the lamina about this axis. The lamina is released from rest with \(A D\) horizontal and \(B C\) below \(A D\).
2. For an instant during the subsequent motion when \(A D\) is vertical, show that the angular speed of the lamina is \(\sqrt { \frac { 3 g } { 50 a } }\) and find its angular acceleration. At an instant when \(A D\) is vertical, the force acting on the lamina at \(A\) has magnitude \(F\).
3. By finding components parallel and perpendicular to \(G A\), or otherwise, show that \(F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }\).
   [0pt] [8]

2. A uniform equilateral triangular lamina \(A B C\) has mass \(m\) and sides of length \(\sqrt { } 3 a\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\), which passes through \(A\) and is perpendicular to the lamina. The midpoint of \(B C\) is \(D\). The lamina is held with \(A D\) making an angle of \(60 ^ { \circ }\) with the upward vertical through \(A\) and released from rest. The moment of inertia of the lamina about the axis \(L\) is \(\frac { 5 m a ^ { 2 } } { 4 }\) Find the speed of \(D\) when \(A D\) is vertical.
(8)

3
\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-2_513_711_890_717} A uniform lamina of mass \(m\) is bounded by concentric circles with centre \(O\) and radii \(a\) and \(2 a\). The lamina is free to rotate about a fixed smooth horizontal axis \(T\) which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about \(T\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). When hanging at rest, with \(O\) vertically below \(T\), the lamina is given an angular speed \(\omega\) about \(T\). The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of \(\omega\).

1. Show that the moment of inertia of the object about \(L\) is \(\left( \frac { 408 + 7 \lambda } { 12 } \right) M a ^ { 2 }\).
   The period of small oscillations of the object about \(L\) is \(5 \pi \sqrt { } \left( \frac { 2 a } { g } \right)\).
2. Find the value of \(\lambda\).

1. Show that the moment of inertia of the body about \(L\) is \(\frac { 77 m a ^ { 2 } } { 4 }\). When \(P R\) is vertical, the body has angular speed \(\omega\) and the centre of the disc strikes a stationary particle of mass \(\frac { 1 } { 2 } \mathrm {~m}\). Given that the particle adheres to the centre of the disc,
2. find, in terms of \(\omega\), the angular speed of the body immediately after the impact.

2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis, the line \(x = 1\) and the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), about the \(x\)-axis. The units are metres, and the density of the solid is \(5400 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). Find the moment of inertia of this solid about the \(x\)-axis.

A uniform disc, of mass \(4 m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2 a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(O A , O B\) and \(O C\), each of mass \(m\) and length \(2 a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42 m a ^ { 2 }\). The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(A O\) making an angle of \(30 ^ { \circ }\) with the horizontal. Find the angular speed of the wheel when \(A O\) is horizontal. When \(A O\) is horizontal the disc becomes detached from the wheel. Find the angle that \(A O\) makes with the horizontal when the wheel first comes to instantaneous rest.

2 A thin rigid rod PQ has length \(2 a\). Its mass per unit length, \(\rho\), is given by \(\rho = k \left( 1 + \frac { x } { 2 a } \right)\) where \(x\) is the distance from P and \(k\) is a positive constant. The mass of the rod is \(M\) and the moment of inertia of the rod about an axis through P perpendicular to PQ is \(I\).

1. Show that \(I = \frac { 14 } { 9 } M a ^ { 2 }\). The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where \(x = \frac { 3 } { 2 } a\). The magnitude of the impulse of this blow is \(J\).
2. Find, in terms of \(a , J\) and \(M\), the initial angular speed of the rod.
3. Find, in terms of \(a , g\) and \(M\), the set of values of \(J\) for which the rod makes complete revolutions.

2 A rod, of length \(x \mathrm {~m}\) and moment of inertia \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude \(J\) Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between \(J , x , I\), the acceleration due to gravity \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and a dimensionless constant \(k\), such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
Find the values of \(\alpha , \beta\) and \(\gamma\) for which this relationship is dimensionally consistent.
[0pt] [6 marks]

4 A uniform circular disc has mass \(m\), radius \(a\) and centre \(C\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(C A = \frac { 1 } { 3 } a\).

1. Find the moment of inertia of the disc about this axis. The disc is released from rest with \(C A\) horizontal.
2. Find the initial angular acceleration of the disc.
3. State the direction of the force acting on the disc at \(A\) immediately after release, and find its magnitude.

1 A circular flywheel of radius 0.3 m , with moment of inertia about its axis \(18 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is rotating freely with angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A tangential force of constant magnitude 48 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

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.