A man of weight \(W\) steadily pulls a sledge of weight \(w\) along level ground by means of a rope (of negligible weight) that passes over his shoulder. The breaking strain in the rope is \(W\), and the relevant coefficients of friction between the man and the ground and between the sledge and the ground are both \(\mu = \tan\lambda\). The rope is inclined to the horizontal at an angle \(\theta\). Calculate the tension in the rope, and prove that the inequalities $$w\sin\lambda \leq W\cos(\theta - \lambda), \quad w\cos(\theta + \lambda) \leq W\cos(\theta - \lambda)$$ must both hold.
(i) If the basic units of mass, length and time are changed in such a way that the measures of these quantities are multiplied by factors \(\mu\), \(\lambda\) and \(\tau\) respectively, what is the effect on the measure of a physical quantity that has dimensions \(M^a L^b T^c\)? Explain why the terms in an equation representing a physical relationship must all have the same dimensions. (ii) Either Show that the transformation of physical measures effected by a change of basic units is an element of a commutative group, explaining the meaning to be given to group multiplication (composition). Or A sailing boat has a critical speed \(V\) that cannot ordinarily be exceeded. Assuming that \(V\) depends only on the density of water \(\rho\), the acceleration due to gravity \((g)\), the length of the boat \((l)\) and the shape of the hull, find for a given shape of hull how \(V\) depends on these parameters.
The motion of a rigid body under given forces is unaffected if the following replacements are made: (a) Any two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) acting at a point \(P\) are replaced by the force \(\mathbf{F}_1 + \mathbf{F}_2\) at \(P\), and vice versa. (b) Any force \(\mathbf{F}\) acting at \(P\) is replaced by a force \(\mathbf{F}\) acting at any other point \(Q\) on the line through \(P\) parallel to \(\mathbf{F}\). If a system of forces can be converted into another by a number of such replacements, the systems are said to be equivalent. Prove (i) that any system of forces lying in a plane is equivalent either to a single force or to a couple (i.e. a pair of forces \(+\mathbf{F}\), \(-\mathbf{F}\)), (ii) that two systems of forces lying in the same plane are equivalent if, and only if, they have the same resultant force and the same moment about any one point.
A pendulum consists of a bob of mass \(M\) suspended by a light string of length \(l\) from a point that is forced to move along a horizontal straight line with displacement \(x(t)\). The air exerts a resistive force on the bob equal to \(kM\) times its speed. Find the exact equation of motion, and show that for small angular deviations \(\theta\) from the vertical it is approximately $$l\ddot{\theta} + kl\dot{\theta} + g\theta = -(\ddot{x} + k\dot{x}).$$ Show that, if \(x = a\cos\omega t\) where \(\omega^2 = g/l\) and \(a\) is small (so that the approximate equation of motion may be used), an (undamped) periodic motion is possible. Determine \(\theta(t)\) for this motion, and calculate the energy dissipated in one complete swing of the pendulum.
Five equal uniform bars, each of mass \(M\), are freely jointed together to form a plane pentagon \(ABCDE\). They are suspended from \(A\), and are constrained by equal light strings \(AC\) and \(AD\) so as to form a regular pentagon. Show without direct calculation that the tension in each string is the same as it would be if the bars were replaced by light rods and a mass \(M\) attached at each vertex. Hence show that this tension has magnitude \(2Mg\cos\frac{1}{5}\pi\).
If a (commutative) ring has multiplicative identity 1, the element \(x\) is said to have order \(n\) if \(n\) is the least positive integer for which \(x^n = 1\). Show, by considering the elements \(-1\) and \(1 + u + u^4\), that, if a ring has an element \(u\) of order 5, then it has either an element of order 2, or one of order 3. [Note: It is possible to have \(1 + 1 = 0\) in a ring.]
The sequence \(a_0, a_1, \ldots, a_{n-1}\) is such that, for each \(i\) \((0 \leq i \leq n-1)\), \(a_i\) is the number of \(i\)'s in the sequence. (Thus for \(n = 4\) we might have \(a_0, a_1, a_2, a_3 = 1, 2, 1, 0\).) If \(n \geq 7\), show that the sequence can only be $$n-4, 2, 1, 0, 0, \ldots, 0, 1, 0, 0, 0.$$ [Hint: Show that the sum of all the terms is \(n\), and that there are \(n - a_0 - 1\) non-zero terms other than \(a_0\), which sum to \(n - a_0\).]
The famous Four Colour Theorem (still unproved) asserts that the regions of any geographical map in the plane may be coloured using only four colours in such a way that regions which touch along an edge are distinctly coloured. In the case when there is a path composed of edges which includes every vertex just once, show that it is possible to colour the map in such a way that two colours are used for the portion enclosed by the path, and two for the remainder. [You may suppose that the regions of the map are straight-edged polygons, whose edges and vertices are called the edges and vertices of the map.]
If \(A\), \(B\), \(C\) are numbers such that \(A t^2 + 2Bt + C \geq 0\) for all real \(t\), show that \(B^2 \leq AC\). By considering \((f(x) + g(x))^2\), show that $$\left(\int_a^b f(x)g(x)dx\right)^2 \leq \int_a^b (f(x))^2 dx \int_a^b (g(x))^2 dx$$ for any continuous functions defined on the interval \([a, b]\). Obtain the inequality $$\int_0^{\pi/2} \sin^4 x \, dx \leq \frac{1}{8}\sqrt{\pi}.$$
The one-player game of Topswaps is played as follows. The player holds a pack of \(n\) cards, numbered from 1 to \(n\) in a random order. If the top card is numbered \(k\), he calls \(k\), reverses the order of the top \(k\) cards, and continues. Show that the pack eventually reaches a constant state in which the top card is numbered 1. [Hint: if \(k > 1\), and, from some point onwards, no card numbered higher than \(k\) is called, then \(k\) is called at most once thereafter.]