Problems

A particle of unit mass moves under the action of a force which is given in polar coordinates \((r, \theta)\) by \begin{equation*} -\frac{4q\cos\theta}{r^3}\hat{\mathbf{r}} - \frac{2q\sin\theta}{r^3}\hat{\boldsymbol{\theta}}, \end{equation*} where \(\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}\) are unit vectors defined in the usual way. The \(r\)-axis is taken to be \(\theta = 0\), and \(q\) is a positive constant. The particle is projected from \(r = a, \theta = 0\), perpendicularly to the \(r\)-axis, with velocity \((8q/a^2)^{\frac{1}{2}}\). Show that in the subsequent motion, \begin{equation*} \left(\frac{dr}{d\theta}\right)^2 = \frac{(r^2-a^2)r^2}{a^2(1+\cos\theta)}. \end{equation*} By using the substitution \(r = a\sec\lambda\), or otherwise, find and sketch the path of the particle.

The Cartesian components of a force which acts on a given particle of unit mass are \((E\cos\alpha t + \dot{y}B, E\sin\alpha t - \dot{x}B)\), where \((x, y)\) is the position of the particle relative to the origin \(O\). \(E\) and \(B\) are positive constants and a dot denotes differentiation with respect to time. The particle is at rest at \(O\) at time \(t = 0\). By introducing the variable \(\omega = x + iy\), or otherwise, find the position of the particle at all future times for any positive value of \(\alpha\). By examination of the solution for small values of \(\alpha\), or otherwise, describe the motion of the particle if \(\alpha = 0\).

Explain briefly the use of the method of complex impedances for solving problems in a.c. electrical networks. What is the relation of the method to that of partial fractions, integral and complementary function in the solution of ordinary differential equations? Find the conditions under which the bridge circuit shown below is in balance (i.e. no current flows through the meter), if the generator has angular frequency \(\omega\).

In a tournament everybody played against everybody else exactly once, and no game ended in a draw. Show that it is possible to order the players in such a way that everybody beat the player coming immediately after him in the ordering. Show also that if no player beat all the others then there are at least three such orderings.

The bus routes in a town have the following properties.

1. [(i)] Any two bus stops are connected by a bus route.
2. [(ii)] Every two different bus routes have exactly one common bus stop.
3. [(iii)] On each bus route there are exactly \(n\) (\(\geq 2\)) bus stops.

Let \(G\) be a group of permutations of a finite set \(X\). Define the stabiliser \(H(\alpha)\) of \(\alpha \in X\) by \[H(\alpha) \equiv \{g:g \in G, g\alpha = \alpha\}.\] The orbit of an element \(\alpha \in X\), \(O(\alpha)\), is the set of elements \(y\) in \(X\) such that \(g\alpha = y\) for some permutation \(g \in G\). Prove the following statements:

1. [(i)] For any \(\alpha \in X\), \(H(\alpha)\) is a subgroup of \(G\).
2. [(ii)] If \(\alpha, \beta \in X\), \(O(\alpha)\) and \(O(\beta)\) either coincide, or they are disjoint.
3. [(iii)] \(|O(\alpha)| \cdot |H(\alpha)| = |G|\), where \(|A|\) denotes the number of elements of a set \(A\).

Let \(n\), \(p\) and \(q\) be integers and suppose that \(1 < p/q < \sqrt[n+1]2\). Prove that \[\sqrt[n+1]2 < \frac{p^n + p^{n-1}q + ... + pq^{n-1} + 2q^n}{p^n + p^{n-1}q + ... + pq^{n-1} + q^n} = r, \quad \text{say},\] and show that \(r\) is a better approximation to \(\sqrt[n+1]2\) than \(p/q\) is.

The vertices \(A\), \(B\), \(C\) of a triangle (which may be assumed not to be right-angled) are given, referred to a suitable origin, by the vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\). Show that the vector positions of the orthocentre \(H\) and the circumcentre \(S\) of the triangle are given by \[(\alpha + \beta + \gamma)\mathbf{h} = \alpha\mathbf{a} + \beta\mathbf{b} + \gamma\mathbf{c},\] and \[2(\alpha + \beta + \gamma)\mathbf{s} = (\beta + \gamma)\mathbf{a} + (\gamma + \alpha)\mathbf{b} + (\alpha + \beta)\mathbf{c},\] where \[\alpha^{-1} = (\mathbf{a} - \mathbf{b})\cdot(\mathbf{a} - \mathbf{c}), \quad \beta^{-1} = (\mathbf{b} - \mathbf{c})\cdot(\mathbf{b} - \mathbf{a}), \quad \gamma^{-1} = (\mathbf{c} - \mathbf{a})\cdot(\mathbf{c} - \mathbf{b}).\] Verify that the centroid of the triangle divides \(SH\) in the ratio \(1:2\).

A convex polyhedron is such that precisely three faces concur in each vertex, and that every face is either a square or an equilateral triangle. Describe the possible cases.

5 points lie within a unit square, or on its boundary. Prove that some pair of them are at a distance apart less than or equal to \(\frac{1}{2}\sqrt{2}\), and that the smallest distance between pairs of points is only equal to \(\frac{1}{2}\sqrt{2}\) in one exceptional case.