Problems

A circular disc of weight \(w\) and radius \(a\) can slide on a smooth vertical rod passing through a small hole at its centre. It is supported horizontally by \(n\) light vertical rods each of length \(2a\) freely hinged to it at equal intervals round its circumference, the upper ends of the rods being freely hinged to fixed supports. An elastic band of modulus \(\lambda\) and natural length \(c\) is placed round the rods. Prove that the equilibrium is stable if \[ w > \lambda n \sin\frac{\pi}{n} \left( \frac{2a}{c} n \sin\frac{\pi}{n} - 1 \right), \] and that otherwise there will be stable equilibrium when the rods are inclined to the vertical at an angle \[ \cos^{-1} \frac{c(w+\lambda n \sin\frac{\pi}{n})}{2a\lambda n^2 \sin^2\frac{\pi}{n}}.\] Assume that the band remains taut throughout and neglect friction.

A heavy elastic particle is projected from a point \(O\) at the foot of an inclined plane of inclination \(\alpha\) to the horizon. The plane through the direction of projection normal to the inclined plane meets the inclined plane in a line \(OA\) which makes an angle \(\phi\) with the line of greatest slope and the direction of projection makes an angle \(\theta\) with \(OA\). Find equations to determine the position of the particle after any number of rebounds and show that the particle will just have ceased to rebound when it again reaches the foot of the plane if \[ \tan\theta\tan\alpha = (1-e)\cos\phi,\] where \(e\) is the coefficient of restitution.

An elastic string has one end fixed at \(A\), passes through a small fixed ring at \(B\) and has a heavy particle attached at the other end. The unstretched length of the string is equal to \(\frac{1}{2}AB\). The particle is projected from any point in any manner. Assuming that it will describe a plane curve, show that the curve is in general an ellipse.

Four particles, each of mass \(m\), are connected by equal inextensible strings of length \(a\) and lie on a table at the corners of a rhombus the sides of which are formed by the strings. One of the particles receives a blow \(P\) along the diagonal outwards. Prove that the angular velocities of the strings after the blow are equal to \(P\sin\alpha/2ma\), where \(2\alpha\) (\(\alpha < \frac{\pi}{4}\)) is the angle of the rhombus at the particle which is struck.

Write a short account of the method of reciprocation, shewing particularly how to reciprocate a circle into a conic of any species. Give some examples shewing the power of the method. \(S\) is the focus of a given conic and a line \(l\) meets the corresponding directrix in \(Z\). \(l'\) is the line joining \(Z\) to the pole of \(l\). A second conic is drawn having \(S\) as one focus and touching \(l, l'\). A common tangent to the two conics touches them at \(Q, Q'\); shew that \(QQ'\) subtends a right angle at \(S\).

\(OX, OY\) are conjugate lines with respect to a fixed conic. \(A\) is any fixed point. A fixed circle through \(O\) and \(A\) cuts \(OX\) in \(P\), and \(AP\) meets \(OY\) in \(Q\). Shew that the locus of \(Q\) is a conic. As a particular case, shew that, if a point moves so that the line joining it to a fixed point is perpendicular to its polar with respect to a conic, the curve traced out is a rectangular hyperbola with its asymptotes parallel to the axes of the original conic. Deduce that four normals can be drawn from any point to an ellipse.

Shew how to sum the series \(a_0+a_1x+\dots+a_nx^n+\dots\), whose coefficients satisfy the relation \(a_n+pa_{n-1}+qa_{n-2}+ra_{n-3}=0\), \(p, q, r\) being given numbers. In the case where \(3a_n-7a_{n-1}+5a_{n-2}-a_{n-3}=0\), and \(a_0=1, a_1=8, a_2=17\), shew that \(2a_n=20n-7+3^{2-n}\).

Shew that

1. [(i)] \(1+2(\cos\alpha + \cos 2\alpha + \dots + \cos n\alpha) = \sin(n+\tfrac{1}{2})\alpha \csc \frac{\alpha}{2}\).
2. [(ii)] \(1+2(\cos\alpha\cos\theta+\cos 2\alpha\cos 2\theta+\dots+\cos n\alpha\cos n\theta) = \frac{\cos n\alpha \cos(n+1)\theta - \cos(n+1)\alpha \cos n\theta}{\cos\theta - \cos\alpha}\).

Deduce that \begin{align*} 1^2\cos\theta + 2^2\cos 2\theta + \dots + n^2\cos n\theta \\ = \tfrac{1}{2}\csc^2\tfrac{1}{2}\theta \{(n+1)^2\cos n\theta - n^2\cos(n+1)\theta - \sin(n+\tfrac{1}{2})\theta\csc\tfrac{1}{2}\theta\}. \end{align*}

Prove that the following definitions of the curvature of a curve at a point \(P\) lead to the same value.

1. [(i)] The rate, in radians per unit of arc, at which the tangent turns.
2. [(ii)] The limit to which the reciprocal of the radius of a circle touching the curve at \(P\) and passing through an adjacent point \(Q\) tends, as \(Q\) tends to \(P\).

Examine the nature of the evolute of a given curve in the neighbourhood of the following points on the given curve

1. [(a)] a point of inflexion,
2. [(b)] a cusp,
3. [(c)] a point of maximum or minimum curvature.

Trace the curve \(b(ay-x^2)=x^3\), and shew that the evolute has a point of inflexion corresponding to the origin of the curve.

Give some account of the theory of a framework of rods, dealing with (i) the number of rods necessary for a frame with \(n\) joints to be just rigid, (ii) the graphical determination of stresses (a) in a 'simple' frame under given forces at the joints, (\(\beta\)) in a non-simple frame. Illustrate by the two cases of a pentagon \(ABCDE\) of rods jointed at the corners, (a) with connecting rods \(AC, CE\), (b) with connecting rods \(AD, CE\), in equilibrium under three given forces at the corners \(B, C, E\).