MI with removed region

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 \(\omega\).

3 A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { } \left( \frac { 7 } { 2 } g a \right)\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(O P\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac { 3 } { 2 } m g ( 1 + 2 \cos \theta )\). Find the speed of \(P\)

1. when it loses contact with the sphere,
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.)
   \(4 \quad A B\) is a diameter of a uniform circular disc \(D\) of mass \(9 m\), radius \(3 a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112 m a ^ { 2 }\). A particle of mass \(3 m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k \sqrt { } ( g a )\). Find the value of \(k\), correct to 3 significant figures.

3 A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { } \left( \frac { 7 } { 2 } g a \right)\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(O P\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac { 3 } { 2 } m g ( 1 + 2 \cos \theta )\). Find the speed of \(P\)

1. when it loses contact with the sphere,
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.)
   \(4 A B\) is a diameter of a uniform circular disc \(D\) of mass \(9 m\), radius \(3 a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112 m a ^ { 2 }\). A particle of mass \(3 m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k \sqrt { } ( g a )\). Find the value of \(k\), correct to 3 significant figures.

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\includegraphics[max width=\textwidth, alt={}, center]{7fcedc6d-8dc1-4159-8a72-be0f6a3f659b-3_355_693_260_726}
\(A B C D\) is a uniform rectangular lamina of mass \(m\) in which \(A B = 4 a\) and \(B C = 2 a\). The lines \(A C\) and \(B D\) intersect at \(O\). The mid-points of \(O A , O B , O C , O D\) are \(E , F , G , H\) respectively. The rectangle \(E F G H\), in which \(E F = 2 a\) and \(F G = a\), is removed from \(A B C D\) (see diagram). The resulting lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane of \(A B C D\). Show that the moment of inertia of this lamina about the axis is \(\frac { 85 } { 16 } m a ^ { 2 }\). The lamina hangs in equilibrium under gravity with \(C\) vertically below \(A\). The point \(C\) is now given a speed \(u\). Given that the lamina performs complete revolutions, show that $$u ^ { 2 } > \frac { 192 \sqrt { } 5 } { 17 } a g .$$

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)

2. \begin{figure}[h] \end{figure} A uniform circular disc has mass \(4 m\), centre \(O\) and radius \(4 a\). The line \(P O Q\) is a diameter of the disc. A circular hole of radius \(2 a\) is made in the disc with the centre of the hole at the point \(R\) on \(P Q\) where \(Q R = 5 a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.

1. Show that the moment of inertia of the lamina about \(L\) is \(69 m a ^ { 2 }\). The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac { 2 \pi } { 3 }\) before it first comes to instantaneous rest,
2. find \(\Omega\) in terms of \(g\) and \(a\).

8. \begin{figure}[h] \end{figure} A uniform circular disc of radius \(2 a\) has centre \(O\). The points \(P , Q , R\) and \(S\) on the disc are the vertices of a square with centre \(O\) and \(O P = a\). Four circular holes, each of radius \(\frac { a } { 2 }\), and with centres \(P , Q , R\) and \(S\), are drilled in the disc to produce the lamina \(L\), shown shaded in Figure 1. The mass of \(L\) is \(M\).

1. Show that the moment of inertia of \(L\) about an axis through \(O\), and perpendicular to the plane of \(L\), is \(\frac { 55 M a ^ { 2 } } { 24 }\) The lamina \(L\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(L\) and which passes through a point \(A\) on the circumference of \(L\). At time \(t , A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta\)
3. Hence find the period of small oscillations of \(L\) about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to \(A O\), of the force exerted on \(L\) by the axis is \(X\).
4. Find \(X\) in terms of \(M , g\) and \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}

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 the centre of the disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
[0pt] [You may assume without proof that the moment of inertia of a uniform hoop of mass \(m\) and radius \(r\) about an axis through its centre and perpendicular to its plane is \(m r ^ { 2 }\).] \begin{figure}[h] \end{figure} A uniform plane shape \(S\) of mass \(M\) is formed by removing a uniform circular disc with centre \(O\) and radius \(a\) from a uniform circular disc with centre \(O\) and radius \(2 a\), as shown in Figure 1. The shape \(S\) is free to rotate about a fixed smooth axis \(L\), which passes through \(O\) and lies in the plane of the shape.
(b) Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 5 } { 4 } M a ^ { 2 }\). The shape \(S\) is at rest in a horizontal plane and is free to rotate about the axis \(L\). A particle of mass \(M\) falls vertically and strikes \(S\) at the point \(A\), where \(O A = \frac { 3 } { 2 } a\) and \(O A\) is perpendicular to \(L\). The particle adheres to \(S\) at \(A\). Immediately before the particle strikes \(S\) the speed of the particle is \(u\).
(c) Find, in terms of \(M\) and \(u\), the loss in kinetic energy due to the impact.