6.04c Composite bodies: centre of mass

3 In this question, give your answers in an exact form.
The region \(R _ { 1 }\) (shown in Fig. 3) is bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 5\), and the curve \(y = \frac { 1 } { x }\) for \(1 \leqslant x \leqslant 5\).

1. A uniform solid of revolution is formed by rotating the region \(R _ { 1 }\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
2. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-4_849_841_735_651} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The region \(R _ { 2 }\) is bounded by the \(y\)-axis, the lines \(y = 1\) and \(y = 5\), and the curve \(y = \frac { 1 } { x }\) for \(\frac { 1 } { 5 } \leqslant x \leqslant 1\). The region \(R _ { 3 }\) is the square with vertices \(( 0,0 ) , ( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\).
3. Write down the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 2 }\).
4. Find the coordinates of the centre of mass of a uniform lamina occupying the region consisting of \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) (shown shaded in Fig. 3).

5.

1. Use integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is \(\frac { 3 } { 4 } h\) from the vertex of the cone.
   (6 marks) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius \(r\) so that the remaining solid consists of a hemisphere of radius \(r\) and a cone of height \(r\) and base radius \(r\) as shown in Figure 3.
2. Find the distance of the centre of mass of the paperweight from its vertex.
   (7 marks)
   

4. \begin{figure}[h] \end{figure} A stand used to reach high shelves in a storeroom is in the shape of a frustum of a cone. It is modelled as a uniform solid formed by removing a right circular cone of height \(2 h\) from a similar cone of height \(3 h\) and base radius \(3 r\) as shown in Figure 1.

1. Show that the centre of mass of the stand is a distance of \(\frac { 33 } { 76 } h\) from its larger plane face.
   (7 marks)
   The stand is stored hanging in equilibrium from a point on the circumference of the larger plane face. Given that \(h = 2 r\),
2. find, correct to the nearest degree, the acute angle which the plane faces of the stand make with the vertical.
   (4 marks)

6. \begin{figure}[h] \end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.

1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
   (7 marks)
   The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
2. Find tan \(\alpha\).
   (6 marks)

6. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
   (3 marks)

5. \begin{figure}[h] \end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.

1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
2. find the value of \(k\).
3. Explain why it may be advantageous to make the base of the flask from a more dense material.
   (2 marks)

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

8. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\). The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. \section*{Advanced Level} \section*{Monday 25 June 2012 - Afternoon} \section*{Materials required for examination
   Mathematical Formulae (Pink)} Items included with question papers
   Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

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 \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).

1. Show that the motion of the rod is approximately simple harmonic.
2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
   (Total 6 marks)

8. Four uniform rods, each of mass \(m\) and length \(2 a\), are joined together at their ends to form a plane rigid square framework \(A B C D\) of side \(2 a\). The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The axis is perpendicular to the plane of the framework.

1. Show that the moment of inertia of the framework about the axis is \(\frac { 40 m a ^ { 2 } } { 3 }\). The framework is slightly disturbed from rest when \(C\) is vertically above \(A\). Find
2. the angular acceleration of the framework when \(A C\) is horizontal,
3. the angular speed of the framework when \(A C\) is horizontal,
4. the magnitude of the force acting on the framework at \(A\) when \(A C\) is horizontal.

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 plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.

1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
   (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
3. 1. Write down an equation of motion for the disc.
   2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.

5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.

1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
2. find \(v\) in terms of \(a\) and \(g\).
3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
   (5)

6.

1. Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through its centre \(O\) perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\). The line \(A B\) is a diameter of the disc and \(P\) is the mid-point of \(O A\). The disc is free to rotate about a fixed smooth horizontal axis \(L\). The axis lies in the plane of the disc, passes through \(P\) and is perpendicular to \(O A\). A particle of mass \(m\) is attached to the disc at \(A\) and a particle of mass \(2 m\) is attached to the disc at \(B\).
2. Show that the moment of inertia of the loaded disc about \(L\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). At time \(t = 0 , P B\) makes a small angle with the downward vertical through \(P\) and the loaded disc is released from rest. By obtaining an equation of motion for the disc and using a suitable approximation,
3. find the time when the loaded disc first comes to instantaneous rest. END

3. A uniform lamina of mass \(m\) is in the shape of a rectangle \(P Q R S\), where \(P Q = 8 a\) and \(Q R = 6 a\).

1. Find the moment of inertia of the lamina about the edge \(P Q\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
   \end{figure} The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge \(P Q\), as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
2. Show that the angular speed of the flap as it reaches the vertical position is \(\sqrt { \left( \frac { g } { 2 a } \right) }\).
3. Find the magnitude of the vertical component of the resultant force of the axis \(P Q\) on the flap, as it reaches the vertical position.

4. A uniform circular disc, of mass \(m\) and radius \(r\), has a diameter \(A B\). The point \(C\) on \(A B\) is such that \(A C = \frac { 1 } { 2 } r\). The disc can rotate freely in a vertical plane about a horizontal axis through \(C\), perpendicular to the plane of the disc. The disc makes small oscillations in a vertical plane about the position of equilibrium in which \(B\) is below \(A\).

1. Show that the motion is approximately simple harmonic.
2. Show that the period of this approximate simple harmonic motion is \(\pi \sqrt { \left( \frac { 6 r } { g } \right) }\). The speed of \(B\) when it is vertically below \(A\) is \(\sqrt { \left( \frac { g r } { 54 } \right) }\). The disc comes to rest when \(C B\) makes an angle \(\alpha\) with the downward vertical.
3. Find an approximate value of \(\alpha\).
   (3)

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

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)

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)