A conic passes through the points \((l, 0), (-l, 0), (0, m), (0, -m)\), where the axes are rectangular. Prove that the locus of the centre of curvature at the point \((l, 0)\) is \[ lm^2\{(l-x)^2+y^2\} + (l-x)\{m^2 y^2 - l^2(l-x)^2\}=0. \]
A rod of constant length moves so that its ends lie on two fixed lines intersecting at right angles. Shew that the motion can be represented by the rolling of a circle (to which the rod is fixed) inside another circle of double the radius, and hence find what points rigidly attached to the rod describe straight lines during the motion.
A uniform heavy rod \(AB\) lying on a rough table has a force applied at the end \(A\) which is gradually increased until the rod is just about to move in such a way that the initial motion will be rotation about a point \(C\) in \(AB\). Prove that the force at \(A\) must be applied at right angles to \(AB\) and that the point \(C\) must divide \(AB\) so that \[ AC:CB = 1+\sqrt{2}:1. \]
A variable conic passes through two fixed points \(A\) and \(B\), and has double contact with a fixed conic \(S\), i.e. touches it at both ends of a variable chord \(l\). Show that the chord of contact, \(l\), passes through one or other of two fixed points \(P\) and \(Q\) which harmonically separate \(A\) and \(B\) and are also conjugate points with respect to \(S\). Determine the coordinates of \(P\) and \(Q\) when \(A\) and \(B\) are the points \((0, 4)\) and \((7, -3)\) and \(S\) is given by the equation \[ 2x^2 + xy + 4y^2 + 6x + 4y + 1 = 0. \]
A Stokes gun is used to fire shells from a point on the same level as the base of a wall and at a distance \(x\) from it. To clear the wall the shells must pass it at a height not less than 96 feet, and to do this the greatest possible value of \(x\) is 280 feet. Taking \(g=32\), shew that the muzzle velocity is 112 feet per second. When \(x\) is less than 280 feet let \(P\) and \(Q\) be the points on the far side of the wall reached by shots which just skim the wall; shew that \(PQ = \frac{192x}{x^2+9216}(78400-x^2)^\frac{1}{2}\). Shew also that, if the greatest possible distance from the gun is not reached by a shot that skims the wall, \(x\) must lie between 224 feet and 168 feet.
A mass \(M\) is hung from a light spring of natural length \(l_1\) and modulus of elasticity \(\lambda_1\) and the other end of the spring is hung from a second light spring (\(l_2, \lambda_2\)), of which the upper end is fixed. Find the equilibrium configuration, and shew that the period of small vertical oscillations is \[ 2\pi \sqrt{M\left(\frac{l_1}{\lambda_1}+\frac{l_2}{\lambda_2}\right)}. \]
When the equation of a line, referred to rectangular axes, is put in the form \(lx + my + 1 = 0\), \((l, m)\) are taken as the coordinates of the line. Interpret the following tangential (line-coordinate) equations: \begin{align*} P_1 &= x_1 l + y_1 m + 1 = 0. \\ \Omega &= l^2 + m^2 = 0. \\ a\Omega - P_1^2 &= 0. \end{align*} Show that if the tangential equation of a conic is \(S=0\), where \[ S=Al^2+2Hlm+Bm^2+2Gl+2Fm+C, \] then the equation of any confocal is \(S + \lambda \Omega = 0\). The foci of a confocal family divide the segment \(AB\) harmonically. Show that there is one member of the confocal family touching the four common tangents of the two circles with centres \(A\) and \(B\), and radii \(a\) and \(b\) respectively, \(a\) and \(b\) being arbitrary.
On a thin smooth wire in the form of a vertical circle of radius \(a\) are two beads of masses \(m\) and \(m'\) respectively which can slide freely on the wire and are connected by a light rod subtending an angle \(2\alpha\) at the centre. Find the angle that the rod makes with the horizontal in stable equilibrium and shew that the time of a small oscillation is \[ 2\pi\left\{ \frac{a(m+m')}{(m^2+m'^2+2mm'\cos 2\alpha)^{\frac{1}{2}}} \right\}^{\frac{1}{2}}. \]
A gun is placed at a height \(H\) above mean sea level and fires at an object on the water at a horizontal distance \(R\) out to sea. Prove that, if \(\delta\) is the angle of depression below the horizontal as seen from the gun, the gun must be elevated at an angle \(\theta\) above the horizontal given approximately by \[ \tan\theta = \frac{gH}{2V^2}\cot\delta - \tan\delta, \] where \(V\) is the muzzle velocity. It is assumed that \(\frac{gR}{V^2}\) and \(\frac{H}{R}\) are small.
Two masses \(m, m'\) lie on a smooth horizontal table connected by a taut unstretched elastic string of modulus of elasticity \(\lambda\) and natural length \(l\). The mass \(m\) is projected with velocity \(v\) in the direction away from the mass \(m'\). Shew that the masses will collide after a time \(\frac{\pi}{a}\frac{l}{v}\), where \(a^2 = \frac{m+m'}{mm'}\frac{\lambda}{l}\).