Small oscillations period

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.

5
\includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.

5
\includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
   The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).

5
\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
   The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).

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
\includegraphics[max width=\textwidth, alt={}, center]{96b6c92d-6d13-452f-84ec-37c45651b232-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find

1. the period of small oscillations,
2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.

5
\includegraphics[max width=\textwidth, alt={}, center]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find

1. the period of small oscillations,
2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.

5
\includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find

1. the period of small oscillations,
2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram).

1. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\).
2. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\).
3. Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).

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.

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
\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.

3
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-2_664_623_904_760} A uniform circular disc has mass \(4 m\), radius \(2 a\) and centre \(O\). The points \(A\) and \(B\) are at opposite ends of a diameter of the disc, and the mid-point of \(O A\) is \(P\). A particle of mass \(m\) is attached to the disc at \(B\). The resulting compound pendulum is in a vertical plane and is free to rotate about a fixed horizontal axis passing through \(P\) and perpendicular to the disc (see diagram). The pendulum makes small oscillations.

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

6 A pendulum consists of a uniform rod \(A B\) of length \(2 a\) and mass \(2 m\) and a particle of mass \(m\) that is attached to the end \(B\). The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\).

1. Show that the moment of inertia of this pendulum about the axis of rotation is \(\frac { 20 } { 3 } m a ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The pendulum is initially held with \(B\) vertically above \(A\) (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is \(\theta\) radians the pendulum has angular speed \(\omega \mathrm { rads } ^ { - 1 }\) (see Fig. 2).
2. Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
3. Find the angular acceleration of the pendulum in terms of \(g , a\) and \(\theta\). At an instant when \(\theta = \frac { 1 } { 3 } \pi\), the force acting on the pendulum at \(A\) has magnitude \(F\).
4. Find \(F\) in terms of \(m\) and \(g\). It is given that \(a = 0.735 \mathrm {~m}\).
5. Show that the time taken for the pendulum to move from the position \(\theta = \frac { 1 } { 6 } \pi\) to the position \(\theta = \frac { 1 } { 3 } \pi\) is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant \(k\). Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}

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.

6. (a) 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 \(B\).
(b) 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,
(c) find the time when the loaded disc first comes to instantaneous rest. 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)

5. \begin{figure}[h] \end{figure} A pendulum \(P\) is modelled as a uniform rod \(A B\), of length \(9 a\) and mass \(m\), rigidly fixed to a uniform circular disc of radius \(a\) and mass \(2 m\). The end \(B\) of the rod is attached to the centre of the disc, and the rod lies in the plane of the disc, as shown in Figure 1. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through the end \(A\) and is perpendicular to the plane of the disc.

1. Show that the moment of inertia of \(P\) about \(L\) is \(190 m a ^ { 2 }\). The pendulum makes small oscillations about \(L\).
2. By writing down an equation of motion for \(P\), find the approximate period of these small oscillations.

5. \begin{figure}[h] \end{figure} A uniform piece of wire \(A B C\), of mass \(2 m\) and length \(4 a\), is bent into two straight equal portions, \(A B\) and \(B C\), which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the wire.

1. Show that the moment of inertia of the wire about \(L\) is \(\frac { 20 m a ^ { 2 } } { 3 }\)
2. By writing down an equation of rotational motion for the wire as it rotates about \(L\), find the period of small oscillations of the wire about its position of stable equilibrium.