A church bell consists of a heavy symmetrical bell and a clapper, both of which can swing freely in a vertical plane about a point \(O\) on a horizontal beam at the apex of the bell. The radius of gyration of the bell (without clapper) about this beam is \(k\) and its centre of mass is at distance \(h\) from \(O\). The clapper may be regarded as a small heavy ball on a light rod of length \(l\). Initially the bell is held with its axis vertical and its mouth above the beam. The clapper rod rests against the side of the bell, making an angle \(\beta\) with the axis. The bell is then released. Show that the clapper rod will remain in contact with the side of the bell until the clapper rod makes an angle \(\alpha\) with the upwards vertical, where \[\cot \alpha = \cot \beta - \frac{k^2}{hl} \textrm{cosec} \beta\] [The radius of gyration, \(k\), is defined by \(I = Mk^2\), where \(I\) is the moment of inertia and \(M\) is the mass.]
A heavy uniform string hangs in a vertical plane over a rough peg which is a horizontal cylinder of circular cross-section whose axis is perpendicular to the plane. The radius of the cylinder is \(a\) and the coefficient of friction is \(\mu\). Let \(T\) be the tension in the string at the point of the cross-section where the tangent makes an angle \(\theta\) with the horizontal. If the string is on the point of slipping in the direction of increasing \(\theta\), show that \[\frac{dT}{d\theta} - \mu T = A(\mu \cos \theta - \sin \theta)\] for a suitable constant \(A\). If one free end lies at the point \(\theta = -\pi/2\), show that the greatest length of string which can hang vertically on the other side of the peg is \(2\mu a (1+e^{\mu\pi})/(1+\mu^2)\).
Two equal smooth perfectly elastic spheres lie at rest on a smooth table, and one is projected so as to strike the other. Show that (unless the impact is direct) the two spheres move at right angles after the impact. Three smooth perfectly elastic spheres of radius \(a\) and equal masses have their centres at the corners \(A\), \(B\), \(C\) of a square, on a smooth table, with \(AB = BC = 2a\) and \(c > 2a\). The sphere at \(A\) is to be projected so as to strike in turn the spheres at \(B\) and \(C\) and finally to move parallel to \(AB\). Show that the direction of projection makes an angle \(\theta\) with \(AB\), where \[a \cos (\theta - \phi) = c \sin \theta \quad \text{and} \quad (c-a)\cos \phi = a.\]
It may be assumed without proof that, in a position of equilibrium of a system, the potential energy has a stationary value; the position of equilibrium is stable when the potential energy is a minimum and unstable when it is a maximum. Three points \(B\), \(A\), \(C\) are in a horizontal line, \(A\) is the midpoint of \(BC\) and \(BC = 2l\). A uniform rod \(AD\), of mass \(M\) and length \(l\), is free to turn about \(A\) in the vertical plane through \(BAC\). Two light strings are attached to the rod at \(D\): one passes through a smooth ring fixed at \(B\) and supports a mass \(m\) which hangs vertically below \(B\); the other passes through a smooth ring fixed at \(C\) and supports an equal mass \(m\) which hangs vertically below \(C\). Show that the potential energy, \(V\), of the system when \(AD\) makes an angle \(\theta\) with the downward vertical is given by the equation \[V = 2\sqrt{2} mgl \cos \frac{1}{2}\theta - \frac{1}{2}Mgl \cos \theta + \text{constant}.\] Prove that there is always at least one position of equilibrium with \(D\) below the line \(BAC\), and that there are three such positions when \(M < 2m < \sqrt{2}M\). Determine for what values of \(M/m\) the position with \(AD\) vertical is stable.
Find the derivative of \(\tan^{-1} [(b^2 - x^2)^{1/2} / (x^2 - a^2)^{1/2}]\) and hence evaluate \[\int_a^b \frac{x\,dx}{(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}\] An unknown function \(f(x)\) is related to a known continuous function \(g(z)\) by \[g(z) = \int_0^z \frac{f(\eta)d\eta}{(z^2-\eta^2)^{1/2}}\] Show that the function \(f(x)\) may be found from \[f(x) = \frac{2}{\pi}\frac{d}{dx}\int_0^x \frac{g(z)z\,dz}{(x^2-z^2)^{1/2}}\]
Let \(R\) be a positive real number. Define a sequence of functions \(V_n(R)\) by \[V_1(R) = 2R,\] \[V_n(R) = \int_{-R}^R V_{n-1}(\sqrt{R^2-x^2})dx, \quad \text{for } n \geq 2.\] Show that \[V_2(R) = \pi R^2,\] \[V_3(R) = \frac{4}{3}\pi R^3,\] and in general \[V_{2n}(R) = \frac{\pi^n R^{2n}}{n!},\] \[V_{2n+1}(R) = \frac{\pi^n 2^{2n+1} n!}{(2n+1)!}R^{2n+1}\] Deduce that \(V_{k+1}(1) < V_k(1)\) for all \(k > 5\), and hence find the maximum value of \(V_k(1)\) for all integers \(k \geq 1\).
Let \(N\), \(r\) be positive integers with greatest common divisor 1, and for each integer \(m \geq 0\) let \(f(m)\) be the remainder on dividing \(r^m\) by \(N\). Show that (i) there exist distinct \(m_1\), \(m_2 > 0\) such that \(f(m_1) = f(m_2)\), (ii) there exists \(m > 0\) such that \(f(m) = 1\). Show that if \(n\) is any integer which is not divisible by 2 or 5, then there is an integer \(k\) such that \(nk\) has all digits 1 when written in base 10.
Let \(b_0\), \(b_1\), \(b_2\), \(b_3\) be integers. Show that \(b_0n^4 + b_1n^3 + b_2n^2 + b_3n\) is divisible by 24 for all integers \(n > 0\) if and only if all of the following conditions are satisfied: (i) \(2b_0 + b_1\) is divisible by 4; (ii) \(b_0 + b_2\) is divisible by 12; (iii) \(b_0 + b_1 + b_2 + b_3\) is divisible by 24.
A set \(S\) of positive integers is called sparse if the equation \(x - y = z - t\) has no solutions with \(x\), \(y\), \(z\), \(t\) in \(S\) apart from those for which \(x = y\) or \(x = z\). Show that the set 1, 2, 4, \ldots of powers of 2 is sparse. Let \(\{u_1, \ldots, u_n\}\) be a sparse set of positive integers, with \(n \geq 2\), and let \(v\) be the smallest positive integer such that \(\{u_1, \ldots, u_n, v\}\) is sparse. Prove that \(v \leq \frac{1}{2}n^3 + 1\). Show that for each integer \(N > 0\) there is a sparse set of positive integers less than or equal to \(N\) containing \([(2N)^{1/3}]\) members. [Here \([X]\) denotes the greatest integer less than or equal to \(X\).]
Express the sum of the fifth powers of the roots of a cubic equation in terms of the sum of the roots, the sum of the squares of the roots and the product of the roots. Prove that \(\frac{(x-y)^5 + (y-z)^5 + (z-x)^5}{(x-y)^2 + (y-z)^2 + (z-x)^2} = \frac{5}{2}(x-y)(y-z)(z-x)\) for all distinct real numbers \(x\), \(y\), \(z\).