Problems

A mass is suspended by a light elastic string from a point \(A\) and produces on extension \(k\), the natural length being \(l\). Prove that if the mass is raised through a vertical distance less than \(k\), and is then let go from rest, it makes oscillations of the same period as a simple pendulum of length \(k\). If the mass is raised up to \(A\) and let fall, show that the maximum extension of the string is \(k(1+\sec\alpha)\), where \(\alpha\) is the acute angle given by \(\tan^2\alpha = \frac{2l}{k}\), and that this extension is attained at a time \((\pi-\alpha)\sqrt{\frac{k}{g}}\) after the string first becomes taut.

Two conics \(S_1, S_2\) cut in \(A, B, C, D\). \(P_1, P_2\) denote the respective poles of \(AB\) and \(CD\) with respect to \(S_1\). \(l_1, l_2\) are two lines through \(P_1, P_2\) respectively. If the pairs of points in which \(l_1\) cuts \(S_1, S_2\) are harmonically conjugate, prove that \(l_2\) is cut harmonically by \(S_1, S_2\). Prove also that \(P_1\) is the pole of \(CD\) with respect to \(S_2\).

\(AB\) is a uniform rod, of length \(6a\) and weight \(W\), which can turn freely about a fixed point in its length distant \(2a\) from \(A\). \(AC\) and \(BC\) are light strings of length \(5a\) attached to a particle \(C\) of weight \(w\). Shew that if \(W\) is less than \(2w\) there will be stable equilibrium with \(AB\) inclined to the horizontal at an angle \(\tan^{-1} \frac{W+w}{4w}\). Point out what are positions of equilibrium when \(W\) is equal to or greater than \(2w\).

Given two tetrahedra \(ABCD, A'B'C'D'\), such that the lines \(AA', BB', CC', DD'\) are concurrent at \(O\), prove that every two corresponding edges, such as \(BC\) and \(B'C'\), intersect each other, and further that the six points of intersection so formed are in one plane, and lie three by three on four lines in that plane.

A right circular cone is circumscribed to a sphere. Shew that, if the radius of the sphere is given, the volume and the total surface area of the cone will each be a minimum when the height of the cone is twice the diameter of the sphere, and that the volume and surface of the cone are then twice those of the sphere.

Prove that \[ \frac{(x-1)(x-2)\dots(x-n)}{x(x+1)(x+2)\dots(x+n)} = \sum_{r=0}^{n} (-1)^{n-r} \frac{(n+r)!}{r! r! (n-r)! (x+r)}, \] and deduce, or otherwise prove, that \[ \frac{(n+1)!}{(n-1)!} - \frac{(n+2)!}{1! 2! (n-2)!} + \frac{(n+3)!}{2!3!(n-3)!} - \frac{(n+4)!}{3!4!(n-4)!} - \dots\dots \] \[ \dots\dots + (-)^n \frac{2n!}{(n-1)! n!} = (-)^n n(n+1). \]

A rhombus consisting of four uniform heavy rods each of length \(l\) jointed together is supported by a fixed vertical circular disc of radius \(r\), one of the corners of the rhombus being vertically above the centre of the disc. Shew that if \(l=10r\) the rods will be inclined to the vertical at an angle \(\tan^{-1} \frac{1}{2}\). Shew also that if \(l=10r+\alpha\), where \(\alpha\) is small, the angle at which the rods are inclined to the vertical will be diminished by \(\frac{36\alpha}{13\pi r}\) degrees approximately.

Prove Chasles' Theorem, namely that if \(ABCD\) are four fixed points on a conic and \(abcd\) the tangents at these points, and if \(P\) is any other point on the conic and \(t\) any other tangent, then the cross-ratio of the pencil \(P(ABCD)\) is constant, and equal to the constant value of the cross-ratio of the range \(t(abcd)\). \(ABC\) is a triangle inscribed in a conic, and \(T\) is the pole of \(AB\). Any line through \(T\) cuts \(BC, AC\) in \(M\) and \(N\). Show that \(M, N\) are conjugate points with respect to the conic.

If \(U=f(r)\), where \(r^2=x_1^2+x_2^2+\dots+x_n^2\) and \(x_1, x_2, \dots, x_n\) are independent variables, prove that \[ \frac{\partial^2 U}{\partial x_1^2} + \frac{\partial^2 U}{\partial x_2^2} + \dots + \frac{\partial^2 U}{\partial x_n^2} = \frac{1}{r^{n-1}}\frac{d}{dr}\{r^{n-1}f'(r)\}. \]

Given that \[ x+iy = \coth \frac{1}{2}(\xi+i\eta), \] express \(x\) and \(y\) separately in real form in terms of \(\xi\) and \(\eta\), and shew that, when \(\xi < 0\), \(y\) can be expressed in a series \[ -2\{e^{\xi}\sin\eta + e^{2\xi}\sin 2\eta + e^{3\xi}\sin 3\eta + \dots\}. \] What is the corresponding series when \(\xi > 0\)?