Problems

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 \(00\). It moves parallel to \(zO\) with velocity \(V\) until its front meets a fixed plane obstacle \(z=0\). Show that the impulsive pressure produced by the collision is \[ \frac{4\rho Va}{\pi^2}\sum_{m=0}^\infty\frac{1}{(2m+1)^2}\sin\frac{(2m+1)\pi x}{a}e^{-\frac{(2m+1)\pi z}{a}}, \] and that the impulse per unit length on the obstacle is \[ \frac{8\rho Va^2}{\pi^3}\sum_{m=0}^\infty\frac{1}{(2m+1)^3}. \]

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.

A Wheatstone bridge has resistances as shown and \(A,B\) are maintained at a constant potential difference \(E\). Show that when \(S\) is large compared with \(R_1\) and \(R_2\), the current through the galvanometer is approximately \[ E(R_2-R_1)/S(R_2+R_1), \] and that the potential difference between \(A\) and \(C\) is approximately \[ ER_1/(R_1+R_2). \] The wires are uniform and made of the same material of specific heat \(\sigma\), and their resistances increase with the temperature \(\theta\) according to the law \[ R_1=r_1(1+\alpha\theta), \quad R_2=r_2(1+\alpha\theta), \] where \(\alpha\) is small. Under the influence of the currents each wire is supposed to heat up uniformly from the temperature \(\theta=0\) without loss of heat. Show that, to the first order in \(\alpha\), \(R_1\) and \(R_2\) increase linearly with the time. If the bridge is initially in balance, show that at a time \(t\) after the potential difference between \(A\) and \(B\) has been applied the current through the galvanometer is approximately \[ \frac{aE^3t}{SJ\sigma(r_1+r_2)^3}\left(\frac{r_2^2}{m_2}-\frac{r_1^2}{m_1}\right), \] where \(J\) is the mechanical equivalent of heat, \(m_1\) the mass of each wire \(AC,DB\) and \(m_2\) the mass of each wire \(AD,CB\).