Two straight rods passing through the fixed points \(A\) and \(B\) revolve uniformly in one plane about these points in the same direction, the one through \(B\) three times as fast as the one through \(A\). The rods, starting in the same direction \(AB\), intersect in \(P\) at any time, and \(N\) is the foot of the perpendicular from \(P\) on \(AB\). Prove that the motion of \(N\) is simple harmonic, with amplitude equal to \(AB\) and period equal to half the time of one revolution of the rod through \(A\).
A mass \(M\) lb. is to be raised through a vertical height \(h\) feet, starting from rest and coming to rest under gravity, by a chain whose tension is not allowed to exceed \(P\) lb. wt. Neglecting the weight of the chain, prove that the shortest time in which this can be done is \[ \sqrt{\frac{2hP}{g(P-M)}} \text{ seconds.} \] If a large amount of material is to be raised, prove that the total time occupied on the upward journeys will be as short as possible if the material is sent up in loads of \(\frac{2}{3}P\) lb.
In a smooth fixed circular tube, of radius \(a\) and small bore, in a vertical plane, are two particles of masses \(m\) and \(2m\), connected by a light inextensible string of length \(a\pi\). With the particles at the ends of the horizontal diameter, and the string in the upper half of the tube, the system is released from rest. Prove that, when each particle has described an arc \(a\theta \left[\theta < \frac{\pi}{2}\right]\), the pressure between the lighter particle \(m\) and the tube is \(\frac{1}{3}mg\sin\theta\), and the tension of the string is \(\frac{2}{3}mg\cos\theta\).
A gun barrel of mass 4 tons is attached to a rigid mounting through a hydraulic buffer, which exerts an opposing force of \(4+\frac{v}{2}\) tons weight when the barrel is recoiling at \(v\) feet per second, the force and the motion both being in the direction of the barrel, which has an elevation of \(30^\circ\). Prove that, when a shot of mass 1 cwt. is discharged at 1280 feet per second, the distance of recoil is 2.4 feet. [\(\log_e 5 = 1 \cdot 6\).]
(i) If \(e^y+e^{-x}=2\), prove that \[ \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0. \] (ii) If the equiangular spiral \(r=ae^{\theta\cot\alpha}\) cuts any straight line through the origin in the consecutive points \(P_1, P_2, \dots, P_n\), and if \(\rho_n\) is the radius of curvature at \(P_n\), prove that \[ \log \frac{\rho_n}{\rho_m} = 2\pi(n-m)\cot\alpha. \]
Within a given circle of radius \(r\) an ellipse is drawn having double contact with the circle, and having one end of its minor axis at the centre of the circle. Prove that the maximum area the ellipse can have is \[ \frac{2\pi r^2}{3\sqrt{3}}. \]
(i) Prove that \[ \int_1^\infty \frac{dx}{x(1+x^3)} = \frac{2}{3}\log_e 2. \] (ii) Find the area of the curve \[ a^2y^2 = x^2(a^2-x^2). \]
A see-saw consists of a plank of weight \(w\) laid across a fixed rough log whose shape is a horizontal circular cylinder. The inclination to the horizontal at which it balances is increased to \(\alpha\) when loads \(W, W'\) are placed at the lower and higher ends respectively: and the inclination is reduced to \(\beta\) when the loads are interchanged. Show that the inclination of the plank when unloaded is \[ \frac{w'(W+W'+w)(W'\alpha - W\beta)}{w(W+W'-w')(W-W')}, \] \(w'\) being the load which, placed at the higher end, would balance the plank horizontal.
Explain the Principle of Virtual Work. A smooth sphere of radius \(r\) and weight \(W\) rests in a horizontal circular hole of radius \(a\). A string is wrapped once round the sphere above the hole and then pulled tight. What tension in the string will just raise the sphere?
A number of small rings can slide freely on a smooth fixed circular wire, and each ring repels every other ring with a force which is measured by the product of their masses and the distance between them: show that, in equilibrium, the centre of gravity of the rings is the centre of the circle.