A 50-ton engine starts from rest with a 10-ton truck: the coupling is initially slack, and when it tightens the engine is running at 2 f.s. Each coupling hook is attached to a spring which extends \(\frac{1}{4}\) inch per ton of pull, the chain itself being inextensible. Find the maximum tension in the chain during the jerk, on the assumption that the engine is running with steam cut off during that time.
Prove that, if \(4x\) lies between \(+1\) and \(-1\), \begin{align*} (1 + \sqrt{1-4x})^4 &= 16 - 64x + 32x^2 - 16x^4 \\ &\quad - 64\left\{x^5 + \frac{7}{2!}x^6 + \frac{8\cdot9}{3!}x^7 + \frac{9\cdot10\cdot11}{4!}x^8 + \dots + \frac{(n+1)(n+2)\cdots(2n-5)}{(n-4)!}x^n + \dots \right\} \end{align*}
A circular disc rests in a vertical plane on a horizontal plane, and in contact with it in the same vertical plane there is a second equal disc which is also in contact with a peg. If all the surfaces in contact are equally rough (\(\mu\)), and the equilibrium is limiting both at the peg and between the two discs, find the position of the peg and the inclination of the line joining the centres of the discs.
Show that if \(ax+by+cz=0\) for all values of \(x, y,\) and \(z\) such that \(\alpha x + \beta y + \gamma z = 0\), then \[ \frac{a}{\alpha} = \frac{b}{\beta} = \frac{c}{\gamma}. \]
The number \(e\) may be defined
A thin uniform metal plate is moving in any manner on a smooth horizontal table; investigate the question whether it can be stopped completely by stopping one point of it, and if so find the point.
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.