A uniform thin hollow right circular cylinder stands upright on a table, and three smooth equal spheres each of weight \(w\) are placed inside it. The ratio of the radius of a sphere to that of the cylinder is \(\alpha\). Prove that if \(\frac{1}{2} > \alpha > 2\sqrt{3}-3\), so that two of the spheres rest upon the ground, the cylinder will not overturn if its weight exceed \(\frac{w}{2}(1+\sqrt{1-2\alpha^2})\). Each sphere is to be taken in contact with the cylinder and with the other two spheres.
Two small rings \(P, Q\) can slide on the upper part of a smooth circular wire in a vertical plane, and are attached by strings of equal length to a third ring \(R\) which is free to slide along the vertical diameter of the circle. The weights of the three rings are equal. Prove that, if the lengths of the strings are less than the radius of the circle, there is a stable position of equilibrium in which \(R\) is at the centroid of the triangle \(POQ\), where \(O\) is the centre of the circle.
A bucket of mass \(m_1\) is joined to a counterpoise of mass \(m_2\) by a light string hanging over a smooth pulley. A ball of mass \(m\) is dropped into the bucket. Shew that the ball will come to rest in the bucket at a time \(\displaystyle\frac{ev(m_1+m_2)}{(1-e)m_2g}\) after the first impact, where \(v\) is the velocity of the ball relative to the bucket immediately before the first impact, and \(e\) is the coefficient of restitution. Shew that the sum of the upward momentum of the system on one side of the pulley and the downward momentum of that on the other side, increases at a uniform rate, and determine this rate. Hence or otherwise shew that the velocity of the system so soon as the ball has come to rest in the bucket is \[ u + \frac{mm_2+e(mm_1+m_1^2-m_2^2)}{(1-e)m_2(m+m_1+m_2)}v, \] where \(u\) is the downward velocity of the bucket immediately before the first impact.
A horizontal rod of mass \(M\) is movable along its length, and its motion is controlled by a light spring which exerts a restoring force \(Ex\) when the rod is displaced through a distance \(x\). A spider of mass \(m\) stands on the rod, and everything is initially at rest. The spider then runs a distance \(a\) along the rod, and then stops, his velocity relative to the rod being constant and equal to \(u\). Shew that the total energy of the system after the run is \[ \frac{2m^2u^2}{M+m}\sin^2\left(\frac{a}{2u}\sqrt{\frac{E}{M+m}}\right), \] and find the amplitude of the final motion.
Write a short account of the method of reciprocation showing particularly how to reciprocate a circle into a conic of any species. Apply the method to the following case: \(S\) 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 having a focus at \(S\) touches \(L, L'\). A common tangent to the conic touches them at \(Q, Q'\). Show that \(QSQ'\) is a right angle.
In the continued fraction \(\displaystyle\frac{1}{a_1+}\frac{1}{a_2+}\dots\), the \(n\)th convergent is denoted by \(p_n/q_n\). Prove that
Prove that if \(A,P,Q\) are polynomials in \(x\) and \(A\) is of lower degree than \(PQ\), then \(A/PQ\) can be expressed in the form \(M/P+N/Q\), where \(M,N\) are respectively of lower degrees than \(P,Q\). Expand \(\displaystyle\frac{\sin\phi}{1-2x\cos\phi+x^2}\) in ascending powers of \(x\), and prove that the remainder after \(n\) terms is equal to \[ \frac{x^n\sin(n+1)\phi-x^{n+1}\sin n\phi}{1-2x\cos\phi+x^2}. \]
Determine the different kinds of conics represented by the equation \[ x^2+4\lambda xy+4y^2+2(1+\lambda)x+8y+5+2\lambda=0, \] for different values of \(\lambda\). Examine especially the critical cases \(\lambda=1, 0, -1, -2\) and illustrate by sketches the transition from one kind of conic to another.
Show that the coordinates of any point on a conic can be expressed in terms of a parameter by the equations \[ \frac{x}{at^2+2bt+c} = \frac{y}{a't^2+2b't+c'} = \frac{1}{a''t^2+2b''t+c''}. \] Find the condition that \(lx+my+n=0\) may be a tangent, and obtain (i) the foci, (ii) the director circle, (iii) the conditions for the conic to be a parabola, or a rectangular hyperbola.
Show that the function \(\sin x+a\sin 3x\) for values of \(x\) between \(0\) and \(\pi\) has two minima with an intermediate maximum if \(a<-\frac{1}{9}\); one maximum if \(-\frac{1}{9}