4 A uniform square lamina has mass \(m\) and sides of length \(2 a\).
- Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane.
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The uniform square lamina has centre \(C\) and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners \(A\). The lamina is initially held such that \(A C\) is vertical with \(C\) above \(A\). The lamina is slightly disturbed from rest from this initial position. When \(A C\) makes an angle \(\theta\) with the upward vertical, the force exerted by the axis on the lamina has components \(X\) parallel to \(A C\) and \(Y\) perpendicular to \(A C\) (see diagram). - Show that the angular speed, \(\omega\), of the lamina satisfies \(a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )\).
- Find \(X\) and \(Y\) in terms of \(m , g\) and \(\theta\).
\section*{Question 5 begins on page 4.}
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A pendulum consists of a uniform rod \(A B\) of length \(4 a\) and mass \(4 m\) and a spherical shell of radius \(a\), mass \(m\) and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the surface of the shell in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\) (see diagram). - Show that the moment of inertia of the pendulum about the axis of rotation is \(47 m a ^ { 2 }\).
A particle of mass \(m\) is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed \(\sqrt { k g a }\), where \(k\) is a positive constant, and strikes the rod at a distance \(3 a\) from \(A\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(P\) starts to rotate.
- Show that the initial angular speed of \(P\) is \(\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }\).
- For the case \(k = 4\), find the angle that \(P\) has turned through when \(P\) first comes to instantaneous rest.
- Find the least value of \(k\) such that the rod reaches the horizontal.
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A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\) is attached to \(A\). The string passes over a small smooth fixed pulley \(C\), where \(A C\) is horizontal and \(A C = a\). The other end of the string is attached to the rod at its mid-point \(D\). The rod makes an angle \(\theta\) below the horizontal (see diagram). - Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by
$$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
- Show that \(\theta = \frac { 1 } { 6 } \pi\) is a position of stable equilibrium for the system.
The system is making small oscillations about the equilibrium position.
- By differentiating the energy equation with respect to time, show that
$$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
- Using the substitution \(\theta = \phi + \frac { 1 } { 6 } \pi\), show that the motion is approximately simple harmonic, and find the approximate period of the oscillations.
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