A room contains \(m\) men and \(w\) women. They leave one by one at random until only persons of the same sex remain. Show by a carefully explained inductive argument, or otherwise, that the expected number of persons remaining is \begin{equation*} \frac{m}{w+1} + \frac{w}{m+1} \end{equation*}
As seen from axes fixed on the rotating earth, a projectile experiences in addition to gravity an additional acceleration \(2\mathbf{t} \times \boldsymbol{\omega}\), where \(\mathbf{t}\) is its velocity and \(\boldsymbol{\omega}\) is the angular velocity of the earth. It may be assumed throughout that \(\omega = |\boldsymbol{\omega}|\) is so small that powers of \(\omega\) of degree \(> 2\) may be neglected. Let \(O\) be a point in the northern hemisphere at latitude \(\lambda \neq \pi/2\). Choose axes with \(Ox\) due south, \(Oy\) due east and \(Oz\) vertical. Show that the equations of motion of the projectile can be written: \begin{align*} \ddot{x} &= 2\omega\dot{y}\sin\lambda, \\ \ddot{y} &= -2\omega(\dot{z}\cos\lambda + \dot{x}\sin\lambda), \\ \ddot{z} &= -g + 2\omega\dot{y}\cos\lambda. \end{align*} A projectile is thrown vertically upwards, reaches maximum height and falls back to ground. Show that the horizontal displacement is opposite in direction and 4 times greater in magnitude than that of a projectile dropped from rest relative to the earth at the same maximum height.
A simple pendulum of mass \(m\) and period \(2\pi/\omega\) is initially at rest. It is then subject to a small horizontal force in the plane of oscillation which builds up linearly from 0 at \(t = 0\) to \(F_0\) at time \(t = T\) and thereafter remains constant. Determine the subsequent motion assuming the oscillations remain small. Show that the maximum possible amplitude of the final motion is \(F_0/m\omega^2\).
An amusing trick is to press a finger down on a marble on a horizontal table top in such a way that the marble is projected along the table with an initial linear velocity \(v_0\) and an initial backward angular velocity \(\omega_0\) about a horizontal axis perpendicular to \(v_0\). The coefficient of sliding friction between the marble and the table is constant, and the radius of the marble is \(a\). For what value of \(v_0/a\omega_0\) does the marble:
Four freely jointed light rods \(AB, BC, CD\) and \(DA\) each have length \(a\). A spring of natural length \(\sqrt{2}a\) joins the points \(B\) and \(D\). A mass is attached at \(C\) and the whole system is suspended in a vertical plane from the point \(A\). When in equilibrium the spring has length \(a\). Show that the period of small vertical oscillations of the mass is \begin{equation*} 2\pi\left(\frac{(\sqrt{6} - \sqrt{3})a}{(4\sqrt{2} - 1)g}\right)^{1/2} \end{equation*}
A spherical water droplet moves in an atmosphere saturated with water vapour. The vapour condenses onto the sphere, increasing the mass at a rate \(\lambda\rho A\), where \(A\) is the surface area of the sphere, \(\rho\) is the density of the water and \(\lambda\) is a constant. Show that the radius of the sphere increases linearly with time. The sphere falls freely and vertically under gravity. Assuming that the vapour particles are at rest before coming into contact with the sphere, show that the sphere will fall with an acceleration which at large times approaches \(\frac{1}{3}g\), where \(g\) is the acceleration due to gravity.
The numbers \(a, b, c, d\) have the property that there exist \(x_1, x_2\), not both zero, such that \begin{align} ax_1 + bx_2 &= 0,\\ cx_1 + dx_2 &= 0. \end{align} Show that there exist numbers \(y_1, y_2\), not both zero, such that \begin{align} ay_1 + cy_2 &= 0,\\ by_1 + dy_2 &= 0. \end{align} [If you use any result about determinants, you should prove it.]
(i) Using integration by parts, or otherwise, show that \begin{align} \int_{0}^{\pi/2} \cos 2x \ln (\textrm{cosec} x) \, dx = \frac{\pi}{4}. \end{align} (ii) Evaluate the integral \begin{align} \int_{0}^{\pi/2} \sin 2x \ln (\textrm{cosec} x) \, dx. \end{align} [You may assume that \(u \ln u \to 0\) as \(u \to 0\).]
The variables \(x\) and \(y\) satisfy the differential equations \begin{align} \frac{dx}{dt} &= 2x + y + e^t,\\ \frac{dy}{dt} &= x + 2y. \end{align} Solve these equations subject to the initial conditions \(x(0) = 0, y(0) = 1\). [You may find it helpful to set \(z = x + \lambda y\) and find the two values of \(\lambda\) such that \(z\) satisfies a first order differential equation which does not explicitly involve \(x\) or \(y\).]
Let \(N = p_1^{a_1} \cdots p_r^{a_r}\), where \(p_1, \ldots, p_r\) are distinct primes and \(a_1, \ldots, a_r\) are positive integers. Find an expression for the number of divisors of \(N\) (including 1 and \(N\)) and show that the sum of these divisors is \begin{align} \prod_{i=1}^{r} \frac{(p_i^{a_i+1}-1)}{(p_i-1)}. \end{align}