Problems

A rough plane is inclined at an angle \(\alpha\) to the horizontal. One end of a light rod is pivoted to a point of the plane, and to the other end of the rod is fastened a mass \(M\) which rests on the plane; the coefficient of friction between \(M\) and the plane is \(\mu\), and the friction between the rod and the plane is negligible. If the rod makes an acute angle \(\beta\) with a line of greatest slope of the plane (both directions being measured up the plane), show that the least horizontal force, acting parallel to the plane, that must be applied to \(M\) in order to prevent slipping is \[ Mg (\sin \alpha \tan \beta - \mu \cos \alpha \sec \beta). \] Find also the thrust in the rod.

Two coplanar circles \(S, T\) have radii 9 and 2 units and their centres are 5 units apart. By inverting \(S\) and \(T\) into concentric circles or otherwise, prove that it is possible to draw six circles in the area between \(S\) and \(T\), each of the six circles touching \(S\) and \(T\) and its two neighbours.

Express \[ y = \frac{4}{(1-x)^2(1-x^2)} \] in partial fractions. Show that, when \(x=0\), the value of \[ \frac{1}{n!} \frac{d^n y}{dx^n} \] is equal to \((n+2)^2\) when \(n\) is even, and \((n+1)(n+3)\) when \(n\) is odd.

Prove that, if \[ \begin{vmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{vmatrix} = 0, \] there exist three numbers \(\xi, \eta, \zeta\), not all zero, and such that \begin{align*} \alpha_1 \xi + \beta_1 \eta + \gamma_1 \zeta &= 0, \\ \alpha_2 \xi + \beta_2 \eta + \gamma_2 \zeta &= 0, \\ \alpha_3 \xi + \beta_3 \eta + \gamma_3 \zeta &= 0. \end{align*} Hence (or otherwise) show that if the nine coefficients \(\alpha_i, \beta_i, \gamma_k\) are all positive and are such that \[ \gamma_1 > \alpha_1 + \beta_1, \quad \beta_2 > \gamma_2 + \alpha_2, \quad \alpha_3 > \beta_3 + \gamma_3, \] then the above determinant cannot vanish.

A uniform lamina in the form of a sector of a circle, of radius \(a\), is bounded by radii that enclose an angle \(2\beta\). Prove that the mass centre is distant \((2a \sin \beta)/3\beta\) from the centre of the circle. \par A wedge-shaped portion of angle \(2\beta\) is cut from a uniform solid sphere of radius \(r\) by two planes which pass through a diameter. Show that the distance of the mass centre of the wedge from this diameter is \[ \frac{3\pi a \sin \beta}{16\beta}. \] (Note: the paper has \(a\) in the formula, but \(r\) is the radius of the sphere. Assuming \(r\) is correct.) \[ \frac{3\pi r \sin \beta}{16\beta}. \]

Two fixed points \(A, B\) lie on a given tangent to a conic \(S\). \(P\) is the pole with regard to \(S\) of a variable line \(p\) through \(A\). Prove that the locus of the point of intersection of \(p\) and \(BP\) is a straight line.

Obtain the quadratic equation whose roots \(\eta\) and \(\bar{\eta}\) are given by \[ \eta = \omega + \omega^3 + \omega^4 + \omega^5 + \omega^9 \quad \text{and} \quad \bar{\eta} = \omega^{-1} + \omega^{-3} + \omega^{-4} + \omega^{-5} + \omega^{-9}, \] where \(\omega = \cos \frac{2\pi}{11} + i \sin \frac{2\pi}{11}\). Deduce that the absolute value of \(\eta\) is \(\sqrt{3}\), and that \[ \sin \frac{2\pi}{11} + \sin \frac{6\pi}{11} + \sin \frac{8\pi}{11} + \sin \frac{10\pi}{11} + \sin \frac{18\pi}{11} = \frac{\sqrt{11}}{2}. \]

The complex variables \(u+iv\) and \(x+iy\) (where \(u,v,x\) and \(y\) are real) are connected by the relation \[ (u+iv)^2 = x+iy. \] Express \(x\) and \(y\) in terms of \(u\) and \(v\); and show that, if the function \(f(x,y)\) becomes on substitution \(\phi(u,v)\), then \begin{align*} x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} &= \frac{1}{2} \left( u \frac{\partial \phi}{\partial u} + v \frac{\partial \phi}{\partial v} \right), \\ y \frac{\partial f}{\partial x} - x \frac{\partial f}{\partial y} &= \frac{1}{2} \left( v \frac{\partial \phi}{\partial u} - u \frac{\partial \phi}{\partial v} \right), \\ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} &= \frac{1}{4} \left( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial v^2} \right) / (u^2+v^2). \end{align*}

A tetrahedron \(ABCD\) is made of six equal uniform smoothly-jointed rods, each of weight \(W\). It is hung freely from the mid-point of \(AB\), so that \(AB\) and \(CD\) are horizontal. Find the thrust in \(CD\).

Prove that, if two ranges \((P, Q, \dots)\), \((P', Q', \dots)\) on different lines \(l, l'\) are homographic, then the locus of the point of intersection of lines \((PQ', P'Q)\) is a straight line. \par If the lines \(l, l'\) are the axes of \(x\) and \(y\) and if the condition that points \(P(x,0)\), \(P'(0,y)\) correspond is \[ axy + bx + cy + d = 0, \] prove that the equation of the locus is \[ bx + cy + d = 0. \]