Show that a uniform chain hangs under gravity in a curve (catenary) with equation that can be written in the form \[ y=c\cosh\frac{x}{c}. \] A uniform chain of length \(2l\) hangs symmetrically over two fixed smooth circular cylinders of equal radii \(r\). The axes of the cylinders are horizontal and parallel and a distance \(2a\) apart on the same level, the chain hanging in a plane perpendicular to the axes. Show that \[ l = \frac{(a-r\sin\theta)(\sec\theta+\tan\theta)}{\log(\sec\theta+\tan\theta)} + r\left(\frac{\pi}{2}-\cos\theta+\theta\right), \] where \(\theta\) is the angle the tangents at the highest points of the catenary make with the horizontal.
The point of suspension \(A\) of a pendulum is caused to move along a horizontal straight line \(OX\). The centre of gravity of the pendulum is \(G\), and \(AG=l\). The radius of gyration about any axis through \(G\) perpendicular to \(AG\) is \(k\). The pendulum can move in the vertical plane containing \(OX\). At time \(t\), \(OA=x\), and the angle between \(AG\) and the vertical is \(\theta\), supposed positive when \(GAX\) is acute. Show that \[ l\cos\theta \frac{d^2x}{dt^2} + (l^2+k^2)\frac{d^2\theta}{dt^2} + lg\sin\theta = 0. \] What condition must \(d^2x/dt^2\) satisfy in order that the pendulum can maintain a constant angle \(\alpha\) to the vertical? Show that, if this condition is maintained, the periodic time of small oscillations about the position is \[ 2\pi\left(\frac{l^2+k^2}{lg\cos\alpha}\right)^{\frac{1}{2}}. \]
A uniform solid cube of side \(2a\) starts from rest and slides down a smooth plane inclined at an angle \(2\tan^{-1}\frac{1}{4}\) to the horizontal, the orientation of the cube being such that its front face is perpendicular to the lines of greatest slope of the plane. The cube meets a fixed horizontal bar placed perpendicular to the direction of motion and at a perpendicular distance \(a/4\) from the plane. Show that, if the cube is to have sufficient velocity to surmount the obstacle when it reaches it, the cube must be allowed first to slide down the plane through a distance greater than \(107a/60\). (The obstacle may be taken to be inelastic and so rough that the cube does not slip on it.)
Explain how Newton's laws of motion enable the concepts of ``mass'' and ``force'' to be defined in terms of observable quantities. Discuss the status of the hypothesis of gravitation in relation to Newton's laws, with particular reference to Newton's choice of an inertial frame of reference.
A die marked with the numbers 1, \dots, 6 is thrown \(r\) times and the \(r\) numbers obtained are added. If the numbers 1, \dots, 6 are equally likely to be obtained on each throw show, by considering the coefficient of \(x^n\) in \((x+x^2+\dots+x^6)^r\), that the probability that the sum of the \(r\) numbers should be \(n\) is \[ \frac{1}{6^r}\left[ \frac{r(r+1)\dots(n-1)}{(n-r)!} - r\frac{r(r+1)\dots(n-7)}{(n-r-6)!} + \frac{r(r-1)}{2!}\frac{r(r+1)\dots(n-13)}{(n-r-12)!} - \dots \right], \] and hence find the probability that the total after four throws should be 14.
The three complex numbers \(z_1, z_2, z_3\) are represented in the Argand diagram by the vertices of a triangle \(Z_1Z_2Z_3\) taken in counterclockwise order. On the sides of \(Z_1Z_2Z_3\) are constructed isosceles triangles \(Z_2Z_3W_1, Z_3Z_1W_2, Z_1Z_2W_3\), lying outside \(Z_1Z_2Z_3\). The angles at \(W_1, W_2, W_3\) all equal \(2\pi/3\). Find the complex numbers represented by \(W_1, W_2, W_3\) and prove that the triangle \(W_1W_2W_3\) is equilateral.
Let \[ \rho = \cos\frac{2\pi}{m} + i\sin\frac{2\pi}{m}, \] where \(m\) is a positive integer. For any integer \(r\) put \[ p_m(r) = \frac{\rho^r}{1-\rho} + \frac{\rho^{2r}}{1-\rho^2} + \dots + \frac{\rho^{(m-1)r}}{1-\rho^{m-1}}. \] By considering the differences \[ p_m(r+1) - p_m(r) \] and the sum \[ p_m(0)+p_m(1)+\dots+p_m(m-1), \] or otherwise, evaluate the \(p_m(r)\) for all \(m\) and \(r\). Show in particular that \[ p_m(0) = \frac{1}{2}(m-1). \]
Two lines \(h\) and \(k\) cut at right angles, \(T\) is a point of their plane, and \(A\) is a fixed point of \(h\). The circle on \(AT\) as diameter meets \(k\) in \(L\) and \(M\), and \(TL, TM\) meet \(h\) in \(L', M'\) respectively. Points \(L'', M''\) are taken on \(L'L, M'M\) respectively, such that \(L'L=LL''\) and \(M'M=MM''\). Show that the triangles \(AL''T\) and \(ATM''\) are similar. If \(T\) is allowed to vary, what is the locus of \(L''\) (or \(M''\))?
The normal to the rectangular hyperbola \(S\) at the point \(P\) cuts \(S\) again in \(N\); the diameter through \(P\) cuts \(S\) again in \(P'\). Prove that \(PP'\) is perpendicular to \(P'N\). The pole of \(PN\) with respect to \(S\) is the point \(Q\); prove that \(QP=QP'\).
A rectangular picture frame hangs from a smooth peg by a string of length \(2a\) whose ends are attached to two points on the upper edge at distances \(c\) from its middle point. Prove that if the depth of the frame exceeds \(2c^2(a^2-c^2)^{-1/2}\) there is no position of equilibrium except that in which the picture frame hangs symmetrically.