One end of a light string is fixed, and the string, hanging vertically in a loop in which a ring of mass \(m\) moves, passes over a smooth fixed pulley and has a mass \(M\) tied to the other end. Shew that in the motion the tension of the string is \(\frac{3Mmg}{4M+m}\).
A particle is describing a circle uniformly; determine the radial force acting on it. \par Two particles are connected by a fine string passing through a smooth ring and describe horizontal circles in the same periodic time; shew that the particles are at the same vertical depth below the ring, and find the ratio in which the string is divided by the ring.
A body is projected from a given point with velocity \(V\), so as to pass through another point at a horizontal distance \(a\) from the point of projection and at a height \(b\) above it. Find an equation to determine the necessary angles of elevation. \par A shot has a range \(c\) on a horizontal plane when the angle of elevation is \(\alpha\) and just reaches the base of a vertical target of height \(2a\), where \(a=c\tan\alpha\). Shew that with the same initial velocity, and with elevation \(\theta+\alpha\), it will strike the target at a distance \(a\sin^2\theta\sec^2(\alpha+\theta)\) below the centre.
If \(y=a+x\log\frac{y}{b}\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) when \(x\) is zero. \par Shew that, if \(x\) is so small that the value of \(x^3\) and of higher powers may be neglected, then \[ y = a+x\log\frac{a}{b} + \frac{x^2}{a}\log\frac{a}{b}. \]
Prove Leibnitz's rule for the repeated differentiation of the product of two functions of \(x\). \par Prove that \[ \left(\frac{d}{dx}\right)^n \frac{\log x}{x} = (-1)^n\frac{n!}{x^{n+1}}\left(\log x - 1 - \frac{1}{2}-\dots-\frac{1}{n}\right). \]
Prove that for a curve, the radius of curvature \(\frac{ds}{d\psi}\) is equal to \[ \left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}} / \frac{d^2y}{dx^2}. \] Prove that the length of the radius of curvature of the curve \(y=\frac{x^3}{3}-x\) is a minimum at points for which \(x=\pm 1.07\), approximately.
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. \par \(S\) 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. \par 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. \par Deduce that four normals can be drawn from any point to an ellipse.
Shew that the coordinates of any point on a conic can be expressed in terms of a parameter \(t\) 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 \(\lambda x + \mu y + \nu = 0\) 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.
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. \par 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 \(2a_n = 20n-7+3^{2-n}\).