Problems

Obtain conditions that the lines \(lx+my+n=0\) and \(l'x+m'y+n'=0\) may be conjugate diameters of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Prove that the equation of the two diameters which are conjugate with respect to each of the conics \[ ax^2+2hxy+by^2=1, \quad a'x^2+2h'xy+b'y^2=1 \] is \[ \begin{vmatrix} y^2 & -xy & x^2 \\ a & h & b \\ a' & h' & b' \end{vmatrix} = 0. \]

Show that the equation \[ \frac{d^n}{dx^n}\left(\frac{1}{x^2+1}\right) = 0 \] has just \(n\) roots (all real), and determine them.

``The principle of virtual work epitomizes the laws of statics.'' State and prove this principle, and discuss the foregoing statement. Illustrate the use of the principle by solving the following problem: \par Two uniform rods AB, BC, of equal lengths and of weights \(W_1, W_2\) respectively, are smoothly hinged together at B and to fixed supports at the same level at A, C, so that the angles CAB and ACB are each equal to \(\beta\). Find the horizontal and vertical components of the action at B.

Three particles A, B, C of the same mass rest on a smooth horizontal table. AB and BC are taut inextensible strings, and the angle ABC is acute and equal to \(\alpha\). A is set in motion with velocity V parallel to CB. Show that when the string again tightens C starts off with velocity \(\dfrac{V}{3+4\tan^2\alpha}\).

If \(\Sigma = 0, \alpha = 0, \beta = 0\) are the tangential equations of a conic and two points, interpret geometrically the equations \[ \Sigma + \alpha\beta = 0, \quad \Sigma + \alpha^2 = 0. \] Two conics \(\Sigma_1\) and \(\Sigma_2\) each have double contact with a third conic \(\Sigma\). Show that two of the points of intersection of the common tangents of \(\Sigma_1\) and \(\Sigma_2\) lie on the line joining the poles of the chords of contact of \(\Sigma_1\) and \(\Sigma_2\) with \(\Sigma\) and form with them a harmonic range.

By considering the points where the curve \[ x^3 + y^3 = axy \] is met by the line \(y=mx\), obtain a parametric representation \(x=\phi(m), y=\psi(m)\) of the curve. \par Show that the three points with parameters \(m_1, m_2, m_3\) are collinear if and only if \[ m_1 m_2 m_3 = -1. \] The tangents at three collinear points \(P_1, P_2, P_3\) of the curve meet the curve again in \(Q_1, Q_2, Q_3\), respectively. Prove that \(Q_1, Q_2, Q_3\) are collinear.

Two light spiral springs, OA, AB, are joined together at A, and particles of equal mass are fastened to the compound spring at A, B respectively; the end O is fixed at a point of a smooth horizontal table. Throughout the movement of the system O, A, B remain in a fixed horizontal straight line, with A between O and B. If the masses oscillate so that the displacements of A, B along OAB at any instant are \(x_1, x_2\) respectively, obtain the equations of motion \begin{align*} \ddot{x_1} + q^2x_1 - p^2x_2 &= 0 \\ \ddot{x_2} - p^2x_1 + p^2x_2 &= 0, \end{align*} where \(2\pi/p\) is the period of oscillation of the mass B when A is held fixed, and \(2\pi/q\) is the period of oscillation of the mass A when B is held fixed with OB equal to the unstretched length of the combined spring. \par For the case in which \(q^2 = \frac{5}{2}p^2\) show that a solution of these equations can be obtained in which \(x_1 = H \cos \sqrt{\frac{1}{3}} pt\), \(x_2 = K \cos \sqrt{\frac{1}{3}} pt\), and find the ratio \(H/K\). \par Show also that a second solution \(x_1 = H' \cos \sqrt{\frac{5}{3}} pt\), \(x_2 = K' \cos \sqrt{\frac{5}{3}} pt\) exists.

A body is free to rotate about a fixed axis. Prove that the rate of change of moment of momentum about the axis is equal to the moment of the external forces about this axis. \par A uniform equilateral triangular lamina of side \(2a\) can rotate freely about a fixed horizontal axis coinciding with one side. Find the length of the equivalent simple pendulum.

A conic circumscribes the triangle ABC and the tangents to it at A, B, C form a triangle PQR. Prove that AP, BQ, CR are concurrent. \par If the conic is a parabola, prove that the point of concurrence lies on the conic touching the sides of the triangle ABC at their middle points.

Establish the formula \[ \frac{1}{2} \int \left(x \frac{dy}{dt} - y \frac{dx}{dt}\right) dt \] for the area of a closed curve given in parametric form \(x=f(t), y=g(t)\), explaining any conventions of sign involved. \par Show that in the parametric representations \[ x = a \cos t, \quad y = b \sin t, \] and \[ x = a \cosh t, \quad y = b \sinh t, \] of the ellipse and the hyperbola, respectively, the area swept out by a radius vector \(OP\) from the centre \(O\), as \(P\) describes an arc of the curve, is proportional to the change in \(t\) along the arc.