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

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)

6. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rod \(A B\), of length \(2 l\) and mass \(4 m\). A particle of mass \(2 m\) is attached to the rod at \(B\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic spring, of natural length \(2 l\) and modulus of elasticity \(k m g\), where \(k > 4\), is attached to the rod at \(B\). The other end of the spring is attached to a fixed point \(C\) which is vertically above \(A\), where \(A C = 2 l\). The angle \(B A C\) is \(2 \theta\), where \(\frac { \pi } { 6 } < \theta \leqslant \frac { \pi } { 2 }\)

1. Show that the potential energy of the system is $$4 m g l \left\{ ( k - 4 ) \sin ^ { 2 } \theta - k \sin \theta \right\} + \text { constant }$$ Given that there is a position of equilibrium with \(\theta \neq \frac { \pi } { 2 }\)
2. show that \(k > 8\) Given that \(k = 10\)
3. determine the stability of this position of equilibrium.

7. \begin{figure}[h] \end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).

1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
2. find the value of \(\theta\) for which the system is in equilibrium,
3. determine the stability of this position of equilibrium.

1. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length \(a\) and modulus \(\frac { 1 } { 4 } m g\), is attached to the end \(B\) of the rod. The other end of the string is attached to a small light smooth ring \(R\). The ring can move freely on a smooth horizontal wire which is fixed at a height \(a\) above \(A\), and in a vertical plane through \(A\). The angle between the rod and the horizontal is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1. Given that the elastic string is vertical,

1. show that the potential energy of the system is $$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
2. Show that when \(\theta = \frac { \pi } { 6 }\) the rod is in stable equilibrium.

7. \begin{figure}[h] \end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.

1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
3. Determine the stability of each position of equilibrium. END

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.

3 A uniform rectangular lamina \(A B C D\) of mass 0.6 kg has sides \(A B = 0.4 \mathrm {~m}\) and \(A D = 0.3 \mathrm {~m}\). The lamina is free to rotate about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.

1. Find the moment of inertia of the lamina about the axis.
2. Find the approximate period of small oscillations in a vertical plane.

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.

5 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 5\) and the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 5\), is occupied by a uniform lamina.

1. Show that the centre of mass of this lamina has \(x\)-coordinate $$\frac { 5 } { 4 } \ln 5 - 1$$
2. Find the \(y\)-coordinate of the centre of mass.

6 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_117_913_251_630} An arm on a fairground ride is modelled as a uniform rod \(A B\), of mass 75 kg and length 7.2 m , with a particle of mass 124 kg attached at \(B\). The arm can rotate about a fixed horizontal axis perpendicular to the rod and passing through the point \(P\) on the rod, where \(A P = 1.2 \mathrm {~m}\).

1. Show that the moment of inertia of the arm about the axis is \(5220 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
2. The arm is released from rest with \(A B\) horizontal, and a frictional couple of constant moment 850 N m opposes the motion. Find the angular speed of the arm when \(B\) is first vertically below \(P\).

2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).

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 \(\frac { 1 } { 7 } m a ^ { 2 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-2_618_627_1594_743} A uniform circular disc has mass 4 kg , radius 0.6 m and centre \(C\). The disc can rotate in a vertical plane about a fixed horizontal axis which is perpendicular to the disc and which passes through the point \(A\) on the disc, where \(A C = 0.4 \mathrm {~m}\). A frictional couple of constant moment 4.8 Nm opposes the motion. The disc is released from rest with \(A C\) horizontal (see diagram).

1. Find the moment of inertia of the disc about the axis through \(A\).
2. Find the angular acceleration of the disc immediately after it is released.
3. Find the angular speed of the disc when \(C\) is first vertically below \(A\).

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).

1. Show that \(\theta = 0\) is a position of stable equilibrium.
2. Find the approximate period of small oscillations about the equilibrium position \(\theta = 0\).

2 \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_462_490_610_834} The diagram shows a uniform lamina \(A B C D E F\) in which all the corners are right angles. The mass of the lamina is \(3 m\).

1. Show that the moment of inertia of the lamina about \(A B\) is \(3 m a ^ { 2 }\).
2. Find the moment of inertia of the lamina about an axis perpendicular to the lamina and passing through \(A\).

3 A uniform rod, of mass 0.75 kg and length 1.6 m , rotates in a vertical plane about a fixed horizontal axis through one end. A frictional couple of constant moment opposes the motion. The rod is released from rest in a horizontal position and, when the rod is first vertical, its angular speed is \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the magnitude of the frictional couple. \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_584_527_1798_822} A disc is rotating about the same axis. The moment of inertia of the disc about the axis is \(0.56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When the rod is vertical, the disc has angular speed \(4.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the opposite direction to that of the rod (see diagram). At this instant the rod hits a magnetic catch \(C\) on the disc and becomes attached to the disc.
2. Find the angular speed of the rod and disc immediately after they have become attached.

5 The region bounded by the \(x\)-axis, the line \(x = 8\) and the curve \(y = x ^ { \frac { 1 } { 3 } }\) for \(0 \leqslant x \leqslant 8\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. The unit of length is the metre, and the density of the solid is \(350 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).

1. Show that the mass of the solid is \(6720 \pi \mathrm {~kg}\).
2. Find the \(x\)-coordinate of the centre of mass of the solid.
3. Find the moment of inertia of the solid about the \(x\)-axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-3_468_550_1201_824} A wheel consists of a uniform circular disc, with centre \(O\), mass 0.08 kg and radius 0.35 m , with a particle \(P\) of mass 0.24 kg attached to a point on the circumference. The wheel is rotating without resistance in a vertical plane about a fixed horizontal axis through \(O\) (see diagram).

1. Find the moment of inertia of the wheel about the axis.
2. Find the distance of the centre of mass of the wheel from the axis. At an instant when \(O P\) is horizontal and the angular speed of the wheel is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
3. the angular acceleration of the wheel,
4. the magnitude of the force acting on the wheel at \(O\).

2 A uniform rectangular lamina has mass \(m\) and sides of length \(3 a\) and \(4 a\), and rotates freely about a fixed horizontal axis. The axis is perpendicular to the lamina and passes through a corner. The lamina makes small oscillations in its own plane, as a compound pendulum.

1. Find the moment of inertia of the lamina about the axis.
2. Find the approximate period of the small oscillations.

3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.

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 \(P\).

2 The region enclosed by the curve \(y = \sqrt { } x\) for \(0 \leqslant x \leqslant 9\), the \(x\)-axis and the line \(x = 9\) is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

3 \includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-2_653_406_727_857} A lamina has mass 1.5 kg . Two perpendicular lines \(A B\) and \(C D\) in the lamina intersect at the point \(X\). The centre of mass, \(G\), of the lamina lies on \(A B\), and \(X G = 0.2 \mathrm {~m}\) (see diagram). The moment of inertia of the lamina about \(A B\) is \(0.02 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the moment of inertia of the lamina about \(C D\) is \(0.12 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis perpendicular to the lamina and passing through \(X\).

1. The lamina makes small oscillations as a compound pendulum. Find the approximate period of these oscillations.
2. The lamina starts at rest with \(G\) vertically below \(X\). A couple of constant moment 3.2 Nm about the axis is now applied to the lamina. Find the angular speed of the lamina when \(X G\) is first horizontal.

5 In this question, \(a\) and \(k\) are positive constants.
The region enclosed by the curve \(y = a \mathrm { e } ^ { - \frac { x } { a } }\) for \(0 \leqslant x \leqslant k a\), the \(x\)-axis, the \(y\)-axis and the line \(x = k a\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of mass \(m\). Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 1 } { 4 } m a ^ { 2 } \left( 1 + \mathrm { e } ^ { - 2 k } \right)\).

2 The region \(R\) is bounded by the curve \(y = \sqrt { 4 a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), the \(x\)-axis, the \(y\)-axis and the line \(x = a\), where \(a\) is a positive constant. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.