Problems

A variable point \(P\) lies in a fixed plane containing a fixed point \(A\). A particle at \(P\) is under the action of a force of magnitude \(\lambda/AP\) (where \(\lambda\) is a constant) directed along \(\overrightarrow{AP}\). Prove that, if the particle is displaced along a straight line from \(P_1\) to \(P_2\), the work done by the force in the displacement is \(\lambda \log\frac{AP_2}{AP_1}\). If \(B\) is another fixed point of the plane, and an additional force of magnitude \(\lambda/BP\) directed along \(\overrightarrow{BP}\) acts on the particle, prove that the work done by the resultant force in the displacement from \(P_1\) to \(P_2\) is \(\lambda \log\frac{AP_2.BP_1}{AP_1.BP_2}\).

Prove that the asymptotes of a rectangular hyperbola bisect the angles between any pair of conjugate diameters. A conic \(\Sigma\) has a focus at the centre of a rectangular hyperbola. A pair of conjugate diameters of the hyperbola meet \(\Sigma\) in \(P, Q, R, S\). Prove that the poles with respect to \(\Sigma\) of \(PR, QR, PS, QS\) lie on the asymptotes of the hyperbola.

If \begin{align*} \cos\theta &= \cos\alpha\cos\phi, \\ \sin(\theta+\phi) &= \lambda\sin\alpha\cos\phi, \\ \text{and prove that} \\ \sin(\theta-\phi) &= \lambda^{-1}\sin\alpha\cos\phi, \end{align*} and hence, or otherwise, express \(\sin\theta\) and \(\sin\phi\) in terms of \(\lambda\) and \(\alpha\). (It is assumed that \(\lambda \sin\alpha, \cos\alpha, \cos\phi\) are all different from 0.)

If \[ I_m = \int_0^{\pi/2} \cos^m x \,dx, \] evaluate \(I_{2n}\) and \(I_{2n+1}\) for all non-negative integers \(n\). Prove that \(I_{2n+2} < I_{2n+1} < I_{2n}\), and deduce that \[ \frac{2.4.6.\dots.2n}{1.3.5.\dots.(2n-1)}\frac{1}{\sqrt{n}} \] tends to the limit \(\sqrt{\pi}\) as \(n\) tends to infinity.

Define the angular velocity of a lamina moving in its own plane. Two circular cylinders \(A, B\) have radii \(a, b\) respectively \((a>b)\) and a common axis. A circular cylinder \(C\) of diameter \(a-b\) touches \(A\) and \(B\), each along a generator. \(A\) rolls on a fixed plane with angular velocity \(\omega\), and \(B\) rotates with angular velocity \(\omega'\) (measured in the same sense). If the surfaces in contact do not slip, show that (i) \(C\) has angular velocity \((a\omega - b\omega')/(a-b)\), and (ii) the plane containing the axes has angular velocity \((a\omega + b\omega')/(a+b)\).

The conic \(S\), the line \(l\) and the point \(A\) are fixed. A variable line \(\lambda\) through \(A\) meets \(l\) in \(P\), and \(Q\) is the point on \(\lambda\) conjugate to \(P\) with regard to \(S\). Prove that the locus of \(Q\) is a conic passing through \(A\), the points of intersection of \(l\) and \(S\), and the pole of \(l\) with regard to \(S\).

Prove that, if \(m\) and \(n\) are fixed positive integers, then \[ \frac{m}{x^m-1} - \frac{n}{x^n-1} \] tends to a limit when \(x\) tends to 1, and find the limit. By putting \(y=x^\lambda\), or otherwise, prove that, if \(\lambda\) is a fixed positive rational number, then \[ \frac{y^\lambda-1}{y-1} \] tends to the limit \(\frac{1}{2}(1-\lambda)\) when \(y\) tends to 1. [The positive value of \(y^\lambda\) for \(y>0\) is to be taken.]

If \(U = f(y/x)\) and \(U_n = r^n U\), where \(r^2 = x^2+y^2\), prove that \[ x\frac{\partial U}{\partial x} + y\frac{\partial U}{\partial y} = 0, \] \[ \frac{\partial^2 U_n}{\partial x^2} + \frac{\partial^2 U_n}{\partial y^2} = r^n\left(\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{n^2}{r^2}U\right). \] Hence, or otherwise, prove that, if the equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] is satisfied by \(u = V(x,y)\), where \(V(x,y)\) is a homogeneous function of degree \(n\), then the equation is satisfied also by \(u = r^{-2n}V(x,y)\).

Two light rods \(AB, BC\), each of length \(a\), are freely jointed at \(B\), and particles of masses \(m_1, m_2, m_3\) are attached at \(A, B, C\) respectively. The system is placed on a rough horizontal turntable, the particles alone making contact with it, so that \(A, B, C\) are at distances \(a, 2a, 3a\) respectively from the centre of rotation. Prove that, if the table rotates with constant angular velocity \(\omega\) and \[ a\omega^2(m_1+2m_2+3m_3) < \mu g(m_1+m_2+m_3), \] where \(\mu\) is the coefficient of friction at each contact, the system can remain upon the table without slipping.

\(A\) is a fixed point on a sphere and \(P\) is a variable point on it. \(AP\) is produced to \(Q\) so that \(PQ\) is of constant length. Prove that the plane through \(Q\) perpendicular to \(PQ\) touches a fixed sphere.