Show that the equation of the curve taken by a uniform chain hanging freely under gravity is of the form \(y=c\cosh x/c\), that the arc-length measured from the vertex is \(c\sinh x/c\), and that the tension at any point is proportional to its height above the directrix. A uniform chain hangs symmetrically with its ends fixed a distance \(4a\) apart and its middle supported at the same level as its ends by a smooth peg. Show that the arrangement is stable provided the length of the chain is less than \(4c\sinh a/c\), where \(c\) is the positive root of the equation \(a/c = \coth a/c\).
Obtain the equations for the free motion of a particle relative to the surface of the rotating earth, near a place whose North latitude is \(\lambda\), in the approximate form \begin{align*} \ddot{x}-2\omega\dot{y}\sin\lambda &= 0, \\ \ddot{y}+2\omega\dot{x}\sin\lambda+2\omega\dot{z}\cos\lambda &= 0, \\ \ddot{z}-2\omega\dot{y}\cos\lambda+g &= 0, \end{align*} where terms of the order \(\omega^2x, \omega^2y, \omega^2z\) are neglected; the axes are rectangular and \(Oz\) is along the upward vertical (the vertical being the direction of apparent gravity), \(Oy\) Easterly and \(Ox\) Southerly; \(\omega\) is the earth's angular velocity and \(g\) the acceleration of apparent gravity. A particle is projected in a Northerly direction at an angle \(\alpha\) with the horizontal. Show that when it again reaches the level of projection it will have been deflected to the East or West according as \(\tan\alpha\) is less than or greater than \(3\tan\lambda\).
A homogeneous circular cone of mass \(M\) rolls with the rim of its base (radius \(R\)) on a rough horizontal plane and its vertex in contact with a fixed vertical cylinder of height \(h\) above the plane. Its axis always passes through a fixed point \(O\) on the axis of the cylinder and the motion is steady, the point of contact \(P\) with the plane describing a circle of radius \(r\) with angular velocity \(\omega\). Show that the reaction \(R\) at the vertex is given by \[ Rb = Mgc - M\omega^2(r-c)(r\cot\theta+a\csc\theta)+(A-C)\omega^2\sin\theta\cos\theta - C\omega^2 r\sin\theta/b, \] where \(c\) is the horizontal projection of \(GP\) (\(G\) being the centre of mass of the cone), \(\theta\) is the inclination of its axis to the vertical, and \(C,A\) are its principal moments at \(O\).
The point of suspension \(O\) of a rigid pendulum is given a very rapid simple harmonic vertical oscillation of period \(2\pi\tau\) and very small amplitude \(a\), regarded as of the same order of smallness as \(\tau\). Find the equation of motion for small deviations in a vertical plane from the position in which the centre of mass \(G\) is vertically above \(O\), and show that it has, to the first order in \(\tau\), a solution of the form \[ \theta=(Ae^{pt}+Be^{-pt})(1+q\tau\cos t/\tau+r\tau\sin t/\tau), \] where \(\theta\) is the inclination of \(OG\) to the vertical, \(p,q,r\) are definite finite constants, and \(A,B\) are the arbitrary constants of integration. Hence show that the vertical position is stable if \(a^2/\tau^2 > 2gl\), where \(g\) is the acceleration due to gravity and \(l\) is the length of the equivalent simple pendulum.
Show that the gravitational potential at a point \(P\) at a distance \(r\) from the centre of mass \(O\) of a gravitating system is approximately, if \(r\) is large compared with the dimensions of the system, \[ \gamma\left(\frac{M}{r}+\frac{A+B+C-3I}{2r^3}\right), \] where \(M\) is the mass of the system, \(A,B,C\) its principal moments of inertia at \(O\), \(I\) its moment of inertia about the line \(OP\), and \(\gamma\) the constant of gravitation. Show that the attraction of a distant particle of unit mass on a homogeneous spheroid of axes \(a,a,c\) produces a couple of magnitude \[ 3\gamma M(a^2-c^2)\sin\theta\cos\theta/5r^3 \] about the diameter perpendicular to the plane containing \(OP\) and the polar axis.
Show that, if \(w=f(x+iy)\), the real and imaginary parts of \(w\) give the velocity potential and stream function in a possible irrotational motion of a liquid in two dimensions. Show that \(w=Az+B/z+C\log z\) solves, with suitable choice of the real constants \(A,B,C\), the problem of a cylinder of radius \(a\) at rest, surrounded by a liquid whose velocity at infinity is \(V\) parallel to \(Ox\), and which has a circulation \(\kappa\) round the cylinder. Find a set of forces which applied to the cylinder and to the liquid will maintain this state of motion.
Deduce from the ordinary equations of motion of an incompressible fluid those of impulsive motion.
At a certain instant a jet of liquid of density \(\rho\) occupies the space specified by \(0
Find the velocity of long waves in a uniform channel of rectangular section containing an incompressible fluid under gravity. Assuming that the tide in a river of depth \(h\) consists of a simple harmonic wave of period \(T\) and that the above theory applies, show that a small floating body will be carried up the river by the flow of the tide a distance \[ \frac{AT}{2\pi}\sqrt{\frac{g}{h}}, \] where \(A\) is the height of the tide. Evaluate numerically for \[ h=18 \text{ feet}, \quad A=2 \text{ feet}, \quad T=12 \text{ hours}. \]
Explain briefly the method of images for the solution of problems in electrostatics. Show that the image of a point charge in a conductor consisting of the outer segments of two orthogonal spheres consists of three point charges, and that the capacity of such a conductor freely charged is \[ a+b-ab/\sqrt{a^2+b^2}, \] where \(a,b\) are the radii of the two spheres.
Show that the mutual potential energy of two small magnets of moments \(M,M'\) is \[ MM'(\cos\epsilon-3\cos\theta\cos\theta')/r^3, \] where \(r\) is the distance apart of their centres, \(\epsilon\) the angle between their axes and \(\theta, \theta'\) the angles made by their axes with the line of centres. Three small magnets of equal moment \(M\) can rotate in a plane about their centres, which are fixed at equal intervals \(a\) along a straight line. Find the periods of the normal oscillations about the position of equilibrium in which the axes all point in the same sense along the line of centres.