Three masses \(m_1, m_2\) and \(m_3\) lie at the points \(A, B\), and \(C\) upon a smooth horizontal table; \(A\) and \(B\), \(B\) and \(C\) are connected by light inextensible strings, and the angle \(ABC\) is obtuse. An impulse \(I\) is applied to the mass \(m_3\) in the direction \(BC\): find the initial velocities of the masses and shew that the mass \(m_2\) begins to move in a direction making an angle \(\theta\) with \(AB\) where \[m_3 \tan\theta + (m_1+m_2)\tan B = 0.\]
Find the equation of the tangent at \((1, 2)\) to the curve given by \[xy(x+y) = x^2+y^2+1,\] and determine the point at which it intersects the curve. Find the asymptotes and trace the curve.
Find the work done in stretching an elastic string. A particle of mass \(m\) lies upon a smooth horizontal table and is attached to three points upon the table, at the vertices of an equilateral triangle of side \(2a\), by means of three strings of natural lengths \(l, l'\) and \(l'\) and of moduli \(\lambda, \lambda'\) and \(\lambda'\) respectively. Shew that if the particle can rest in equilibrium at the centre of the triangle, then \[2a(\lambda/l - \lambda'/l') = (\lambda-\lambda')\sqrt{3}.\] Find also the period of a small oscillation of the particle in the line of the string of natural length \(l\).
A rectangular plate \(2a\) by \(2b\) is of thickness \(kr^2\), where \(r\) is the distance of any point from a point distant \(c\) from the centre of the plate. Shew that the mean thickness of the plate is \(k\left(\frac{a^2+b^2}{3} + c^2\right)\).
Integrate \[\int \tan^3\theta d\theta, \quad \int \frac{dx}{x^4+1}, \quad \int \frac{d\theta}{a \sin\theta + b \cos\theta}.\]
Three exactly similar books each of length \(l\) and of uniform density along their length lie in a heap upon a table with their backs in a vertical plane. Each book extends beyond the book below it by a length \(a\). Find the greatest value of \(a\) consistent with equilibrium (i) when the end of the top book just rests on a smooth horizontal ledge (the book remaining horizontal); and (ii) when the top book has no support other than the book below it.
The framework of freely jointed light rods \(ABCD\) supports a weight \(W\) at \(D\) and is freely hinged on to a vertical wall at \(A\) and \(B\). \emph{[A diagram shows a framework ABCD hinged at A and B to a vertical wall, with B above A. A weight W hangs from D. The angles are given as: \(\angle ABC = 90^\circ\); the angle between the downward vertical from A and the rod AC is \(30^\circ\); \(\angle CAD = 30^\circ\); and the angle at D between CD and a line parallel to BC is \(90^\circ\).]} By means of a force diagram or otherwise find the magnitude and direction of the reactions at \(A, B\) and the thrust and tension in each rod.
State Hooke's Law connecting the tension in an elastic string and its extension. A weight \(W\) is suspended by two strings, each of natural length 36 inches, from two points 36 inches apart. The strings have different coefficients of elasticity and are stretched by the weight to lengths 37 and 38 inches. Prove that the coefficient of elasticity of the one string is about 2.2 times that of the other.
A uniform heavy horizontal beam is supported at its two ends \(A, B\) and carries a weight \(W\) at \(C\), where \(AC=2CB\). Shew that, if an ordinate \(y\) is drawn proportional to the bending moment at \(P\) in the rod, where \(AP=x\), then the curve obtained consists of portions of two equal parabolas with axes vertical. If the weight of the beam is also \(W\) and if \(AB=l\), find the latus rectum of the parabolas.
A uniform chain is suspended from one end and the other end hangs over a rough pulley. Prove that the friction brought into play at the pulley is the weight of a length of chain which would reach from the loose end to the directrix of the catenary formed by the chain.