Prove that if three segments \(AB\), \(BC\), \(CD\) of a straight line subtend the same angle \(\theta\) at a point \(P\), \[ 4 \cos^2 \theta = \frac{AC}{CD} \div \frac{AB}{BD}. \] Prove also that, if \(A\), \(B\), \(C\), \(D\) are four points in order upon a straight line, and if \[ AC.BD < 4AB.CD, \] a point \(P\) can be found at which the segments \(AB\), \(BC\), \(CD\) subtend the same angle. Give a construction to determine \(P\).
Two ships are steaming along straight courses which converge at an angle of \(60^\circ\). If their distances from the point of convergence are 20 and 12 nautical miles, and their speeds 16 and 8 knots, respectively, draw a diagram shewing their positions when they are at a distance of 4 miles apart, and find their shortest distance apart.
Assuming the binomial theorem for a positive or negative integral exponent, show that the coefficient of \(x^n\) in the expansion of \[ (1+x+x^2+\dots+x^{n-1})^p, \] where \(p\) is a positive integer, is \[ \frac{(p+n-1)!}{(p-1)!n!}p. \]
Write an account of the theory of plane frames formed of light rigid bars, freely jointed, considering the relation between the number of joints and bars for a ``just stiff'' framework, and describing the method of reciprocal diagrams for determining the stresses in such a frame due to given external forces. \par Explain Bow's notation. \par Illustrate your remarks by considering the frame % Image of a Warren truss with two downward forces % Diagram shows a truss with 5 joints on the bottom chord, 4 on the top. % A single load W is at the second bottom joint from the left. % A load 2W is at the fourth bottom joint from the left. % The truss is supported at the two ends of the bottom chord. % For transcription purposes, the diagram is described rather than drawn.
It is desired that the performance of a model of a machine should correspond with that of the machine itself. Explain how its masses, linear dimensions, and speed, and the forces applied to it, must be adjusted in comparison with those of the machine.
Solve the equations
Two particles of mass \(M\) and \(m\) (\(M>m\)) are placed on the two smooth faces of a light wedge which rests on a smooth horizontal plane. The faces of the wedge are inclined to the horizontal at angles \(\alpha\) and \(\beta\), respectively. If the system starts from rest, shew that the smaller particle will move up the face on which it is placed if \[ \tan \beta < \frac{M \sin\alpha \cos\alpha}{M \sin^2\alpha+m}. \]
Solve the equations:
State the principle of virtual work, and give a proof of it for the case of a single rigid body. What types of force do not appear in the equation of virtual work? Explain how the work function may be used to determine the stability of a system. \par Two equal particles repel each other according to the fifth power of the distance, and are connected by an elastic string. Find the position of equilibrium, and shew that it is stable if the extension of the string is less than one-quarter of its original length.
A frame of steel bars, in the form of a square and two diagonals, is suspended by one angle, a given weight being attached to the opposite angle. The sides of the square and the horizontal diagonal are of equal thickness, half that of the vertical diagonal. Find the tension of the vertical diagonal, neglecting the weight of the frame, and assuming that it has no stress when not loaded.