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The point of suspension \(A\) of a pendulum is caused to move along a horizontal straight line \(OX\). The centre of gravity of the pendulum is \(G\), and \(AG=l\). The radius of gyration about any axis through \(G\) perpendicular to \(AG\) is \(k\). The pendulum can move in the vertical plane containing \(OX\). At time \(t\), \(OA=x\), and the angle between \(AG\) and the vertical is \(\theta\), supposed positive when \(GAX\) is acute. Show that \[ l\cos\theta \frac{d^2x}{dt^2} + (l^2+k^2)\frac{d^2\theta}{dt^2} + lg\sin\theta = 0. \] What condition must \(d^2x/dt^2\) satisfy in order that the pendulum can maintain a constant angle \(\alpha\) to the vertical? Show that, if this condition is maintained, the periodic time of small oscillations about the position is \[ 2\pi\left(\frac{l^2+k^2}{lg\cos\alpha}\right)^{\frac{1}{2}}. \]

Shew that the area of the surface of the prolate spheroid obtained by the rotation of an ellipse of eccentricity \(e\) about its major axis (\(2a\)) is \[ A = 2\pi a^2\left[ \sqrt{1-e^2} + \frac{\sin^{-1} e}{e} \right] \] and that the centroid of the half surface bounded by the central circular section is at a distance \(d\) from the plane of that section, where \[ Ad = \frac{4\pi a^3}{3} \frac{1}{e^2}\left[ \sqrt{1-e^2} - (1-e^2)^{\frac{3}{2}} \right]. \]