Energy method angular speed

5
\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-3_319_794_255_678} A uniform rod \(A B\), of mass \(m\) and length \(6 a\), is rigidly attached at \(B\) to a point on the circumference of a uniform circular lamina of mass \(m\), radius \(2 a\) and centre \(O\). The lamina and the rod are in the same vertical plane, and \(A B O\) is a straight line (see diagram). Show that the moment of inertia of the system about an axis \(l\) through \(A\) perpendicular to the plane of the lamina is \(78 m a ^ { 2 }\). A particle of mass \(2 m\) is now attached at \(B\) and the system is free to rotate in a vertical plane about the fixed axis \(l\) which is horizontal. Initially \(A B\) is horizontal, with \(O\) moving downwards and the system having angular velocity \(\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)\). At time \(t , A B\) makes an angle \(\theta\) with the downward vertical through \(A\).

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }\).
2. Find the angular velocity of the system when \(B\) is vertically below \(A\).

5 Four identical uniform rods, each of mass \(m\) and length \(2 a\), are rigidly joined to form a square frame \(A B C D\). Show that the moment of inertia of the frame about an axis through \(A\) perpendicular to the plane of the frame is \(\frac { 40 } { 3 } m a ^ { 2 }\). The frame is suspended from \(A\) and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through \(A\). When the frame is at rest with \(C\) vertically below \(A\), it is given an angular velocity \(\sqrt { } \left( \frac { 6 g } { 5 a } \right)\). Find the angular velocity of the frame when \(A C\) makes an angle \(\theta\) with the downward vertical through \(A\). When \(A C\) is horizontal, the speed of \(C\) 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]{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 \(P\).

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 \(A\).

5 A uniform rectangular lamina \(A B C D\) has mass 20 kg and sides of lengths \(A B = 0.6 \mathrm {~m}\) and \(B C = 1.8 \mathrm {~m}\). It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(A B\).

1. Show that the moment of inertia of the lamina about the axis is \(22.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_442_541_477_800} The lamina is released from rest with \(B C\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 Nm about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).
2. Show that the angular acceleration is zero when \(\cos \theta = 0.25\).
3. Hence find the maximum angular speed of the lamina.
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_633_838_1356_650} A ship \(P\) is moving with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A second ship \(Q\) is moving with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of \(050 ^ { \circ }\) (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m .
4. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), to the nearest degree, and find the bearing of the other possible direction of this relative velocity. Given that the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), find
5. the bearing of the direction in which \(Q\) is moving,
6. the magnitude of the velocity of \(Q\) relative to \(P\),
7. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together,
8. the bearing of \(P\) from \(Q\) when the two ships are closest together.
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-4_560_1180_265_467} A uniform rod \(A B\) has mass \(m\) and length \(6 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(A C = a\). The angle between \(A B\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(A B\) and \(S\) perpendicular to \(A B\) (see diagram). The rod is released from rest in the position where \(\theta = \frac { 1 } { 3 } \pi\). Air resistance may be neglected.
9. Find the angular acceleration of the rod in terms of \(a , g\) and \(\theta\).
10. Show that the angular speed of the rod is \(\sqrt { \frac { 2 g ( 1 - 2 \cos \theta ) } { 7 a } }\).
11. Find \(R\) and \(S\) in terms of \(m , g\) and \(\theta\).
12. When \(\cos \theta = \frac { 1 } { 3 }\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude.

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\includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-2_392_746_908_645} A non-uniform rectangular lamina \(A B C D\) has mass 6 kg . The centre of mass \(G\) of the lamina is 0.8 m from the side \(A D\) and 0.5 m from the side \(A B\) (see diagram). The moment of inertia of the lamina about \(A D\) is \(6.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\) and the moment of inertia of the lamina about \(A B\) is \(2.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina rotates in a vertical plane about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.

1. Write down the moment of inertia of the lamina about this axis. The lamina is released from rest in the position where \(A B\) and \(D C\) are horizontal and \(D C\) is above \(A B\). A frictional couple of constant moment opposes the motion. When \(A B\) is first vertical, the angular speed of the lamina is \(2.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the moment of the frictional couple.
3. Find the angular acceleration of the lamina immediately after it is released.

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.
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