\(PA_1A_2...A_{2n}Q\) is the chain of a suspension bridge. Each of the vertical bars \(A_1B_1, A_2B_2,...A_{2n}B_{2n}\) bears an equal portion of the weight of the roadway. The distances \(B_0B_1, B_1B_2, ... B_{2n}B_{2n+1}\) are all equal. The weights of the chain and bars may be neglected in comparison with the weight of the roadway. By means of a force diagram, or otherwise, shew that the points \(P, A_1, A_2, ... A_{2n}, Q\) lie on a parabola whose axis is vertical. If \(W\) is the total weight of roadway supported by the bars \(A_1B_1, ... A_{2n}B_{2n}\), \(d\) the depth of \(A_nA_{n+1}\) below \(PQ\), and \(l\) the total span of the bridge, shew that the tension in the chain at \(P\) or \(Q\) is \[ \frac{W}{2}\sqrt{1+\frac{(n+1)^2l^2}{4(2n+1)^2d^2}}. \]
The end \(P\) of a straight rod \(PQ\) describes with uniform angular velocity a circle whose centre is \(O\), while the other end \(Q\) moves on a fixed line through \(O\) in the plane of the circle. The end \(Q'\) of an equal straight rod \(PQ'\) moves on the same fixed line through \(O\). Prove that the velocities of \(Q\) and \(Q'\) are in the ratio \(QO:OQ'\).
A battleship is steaming ahead with velocity \(V\). A gun is mounted on the battleship so as to point straight backwards, and is set at an angle of elevation \(\alpha\). If \(v\) is the velocity of projection (relative to the gun), shew that the range is \(\displaystyle\frac{2v}{g}\sin\alpha(v\cos\alpha-V)\); also shew that the angle of elevation for maximum range is \(\cos^{-1}\{(V+\sqrt{V^2+8v^2})/4v\}\).
A particle moves under a force directed towards a fixed point \(O\). Shew that its path lies in a plane and that \(pv\) is constant, where \(v\) is the velocity of the particle at any instant and \(p\) the length of the perpendicular from \(O\) to the tangent to the path. A particle is repelled from a centre of force \(O\) with a force \(\mu r\) per unit mass, where \(r\) is the distance of the particle from \(O\). Shew that, if the particle is projected from a point \(P\) in any direction with velocity \(OP\sqrt{\mu}\), its path is a rectangular hyperbola with \(O\) as centre.
A railway truck is at rest at the foot of an incline of 1 in 70. A second railway truck of equal weight starts from rest at a point 1000 feet up the incline, and runs down under gravity. The trucks collide at the foot of the incline, the coefficient of restitution being \(\frac{1}{3}\). Find how far each truck travels along the level, the frictional resistances for each truck being 16 lbs. wt. per ton, both on the incline and on the level. Where the incline meets the level, the rails are slightly curved, each in a vertical plane, so that there is no vertical impact, and at the instant of collision both trucks are on the level. (Assume that \(g=32\) ft. sec.\(^{-2}\))
State Hooke's law. A mass \(m\) hangs from a fixed point by means of a light spring, which obeys Hooke's law. Shew that, if the mass be given a small vertical displacement, the ensuing motion of the mass is simple harmonic. If \(n\) is the number of oscillations per second in this simple harmonic motion, and if \(l\) is the length of the spring when the system is in equilibrium, find the natural length of the spring, and shew that, when the spring is extended to double its natural length, the tension is \(m(4\pi^2n^2l-g)\).
A rope hangs over a pulley, whose moment of inertia is \(I\), and which is perfectly smooth on its bearings, but perfectly rough to the rope. Two monkeys of equal mass \(m\) hang one on each end of the rope. The monkeys can climb with constant speeds \(u_1\) and \(u_2\) relative to the rope (\(u_1 > u_2\)). Shew that in a race through a height \(h\) the monkey of speed \(u_1\) can give the other monkey any start up to \[ hI(u_1-u_2)/\{(I+ma^2)u_1+ma^2u_2\}, \] where \(a\) is the radius of the pulley. (The system is at rest before the monkeys start climbing.)
A series of circles touch a given straight line at a given point. Show that the middle points of the chords of contact of the tangents from a fixed point all lie on a circle.
State and prove the property from which the nine points circle of a triangle derives its name. \(RS\) and \(PQ\) are the diameters of the nine-points circle and the circumcircle of the triangle \(ABC\) which are at right angles to the side \(BC\) and \(E\) is the middle point of \(BC\); prove that the lines \(ER, ES, AP, AQ\) form a rectangle.
Define the polar of a point with respect to a circle and show that a straight line through a point cutting a circle is divided harmonically by the circle, the point and the polar of the point. \(TP, TQ\) are tangents to a circle. The perpendicular from \(T\) on any diameter \(AB\) cuts that diameter in \(X\). Prove that \(PX.XQ=AX.XB\).