Two weights \(W_1\) and \(W_2\) are attached to the ends of a rope (of negligible weight) which is passed over a fixed rough horizontal cylinder of circular cross-section. By considering the forces on an element of rope in contact with the cylinder when the friction is limiting, find the manner in which the tension in the rope varies along its length, and hence show that static equilibrium is possible only if \[e^{-\mu\pi} \leq W_1/W_2 \leq e^{\mu\pi},\] where \(\mu\) is the static coefficient of friction between rope and cylinder. How is this result modified if the rope is coiled round the cylinder \(n\) times, the weights \(W_1\) and \(W_2\) being still suspended from its ends?
A mouse runs along a straight line \(y = 0\) with uniform speed \(V_1\). When the mouse is at the point \(x = 0\), \(y = 0\) it is spotted by a cat at the point \(x = 0\), \(y = b\) which immediately gives pursuit. The cat runs with constant speed \(V_2\) (\(> V_1\)) and is always directed at the fleeing mouse. Make a qualitative sketch of the path of the cat (i) relative to a fixed frame of reference, and (ii) relative to a frame of reference moving with the mouse. Let \((r, \theta)\) be polar coordinates in this latter frame of reference, the mouse being at \(r = 0\) and \(\theta\) being measured from its direction of motion. Show that the differential equation of the path of the cat is \[\frac{1}{r}\frac{dr}{d\theta} = -\frac{V_2}{V_1}\csc\theta-\cot\theta.\] Integrate this, and show that if \(V_2 = 2V_1\) the cat catches the mouse after a time of pursuit \(2b/3V_1\).
A rectangular sheet of adhesive material is placed with its adhesive side uppermost on a plane which is inclined to the horizontal at an angle \(\alpha\). Two opposite edges of the sheet (of length \(b\)) are horizontal. The coefficient of friction between the sheet and the plane is sufficient to prevent slipping. The material has uniform mass per unit area \(m\), and its thickness is negligible. A uniform solid cylinder of mass \(M\), radius \(a\) and length \(b\) is placed along the top edge of the material and released from rest. The material then adheres to the cylinder as it rolls downwards. Show that when the cylinder has rolled through an angle \(\theta\), and an area \(ab\theta\) of material has been lifted from the plane, the potential energy of the system has decreased by an amount \[Mga\theta\sin\alpha - mga^2b[(\theta - \sin\theta)\cos\alpha + (1 - \frac{1}{2}\theta^2 - \cos\theta)\sin\alpha].\] By examining the approximate form of this expression when \(\theta\) is small, find an approximation to a value of \(\theta\) for which equilibrium is possible when \(M\tan\alpha\) is small compared with \(mab\).
A horizontal platform is free to rotate about a smooth vertical axis, \(I\) being its moment of inertia. A man of mass \(M\) can walk on the platform. Initially, the man and the platform are at rest. The man walks along a path \(x(t)\), \(y(t)\), where \((x, y)\) are Cartesian coordinates in axes fixed on the platform, and \(t\) is the time. What is the resultant angular velocity \(\dot{\theta}(t)\) of the platform? If the man completes a simple closed path and his maximum distance \(a\) from the centre is such that \(Ma^2\) is much smaller than \(I\), show that the total angle turned by the platform is approximately \(2MA/I\), where \(A\) is the area enclosed by his path.
It is desired to write a computer program that will print out all the prime numbers 2, 3, 5, ... less than some specified \(N\), in their natural order. Give a flow diagram, with any additional explanation you wish to include, for such a program. [More credit will be given for more efficient schemes.]
(i) Show that \(\sum_{r=0}^{n} \binom{n}{r} = 2^n\) for each positive integer \(n\), where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). (ii) Show that, for all positive integers \(r\) and \(n\), \(\sum_{s=0}^{n} \binom{r+s}{r} = \binom{n+r+1}{r+1}\).
(i) Prove that 24 is the largest integer divisible by the product of all integers less than its square root. (ii) Show that in any set of \(n + 1\) numbers chosen from \(1, 2, ..., 2n\), there is always a pair of different numbers, one of which divides the other.
Let \(a\) be a positive integer, and write \(r = \sqrt{a} + \sqrt{(a+1)}\). Show, for each positive integer \(n\), that \(a_n = \frac{1}{4}(r^{2n} + r^{-2n} - 2)\) is an integer, and that \(r^n = \sqrt{a_n} + \sqrt{(a_n + 1)}\). [Positive square roots are to be taken throughout.]
Show that, if \(p = \cos A + \cos B\) and \(q = \sin A + \sin B\), then \(\sin(A + B) = \frac{2pq}{p^2+q^2}\) and \(\cos(A + B) = \frac{p^2-q^2}{p^2+q^2}\). Hence, or otherwise, find the general solution of the equation \(\frac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta} = \frac{\sqrt{2}-2\sin\theta}{\sqrt{2}+2\cos\theta}\).
A theorem in combinatorial theory may be stated as follows: Let \(G_1, G_2, ..., G_n\) be \(n\) girls and \(B_1, B_2, ..., B_n\) \(n\) boys. In order for all boys to be able to choose as dancing partners girls with whom they are friendly, it is necessary and sufficient that, for each subset \(S\) of boys, the number of girls friendly with at least one boy in \(S\) is at least equal to the number of boys in \(S\). Prove the necessity of the condition on subsets of boys, and establish its sufficiency for \(n \leq 3\). Find the number of ways in which dancing partners may be chosen so that only friendly couples dance together, in the following situation with \(n = 5\): \begin{align*} G_1 \text{ is friendly only with } B_1, B_2, B_3 \text{ and } B_4\\ G_2 \text{ is friendly only with } B_1, B_2 \text{ and } B_5\\ G_3 \text{ is friendly only with } B_4 \text{ and } B_5\\ G_4 \text{ is friendly only with } B_1, B_2 \text{ and } B_4 \text{, and}\\ G_5 \text{ is friendly only with } B_3 \end{align*}