Obtain Euler's equations for the motion of a rigid body about a fixed point in the form \[ A\dot\omega_1 - (B-C)\omega_2\omega_3 = L, \] and two similar equations. Show that \((B-C)\omega_2\omega_3\) is equal to the sum of the moments round the axis \(O\xi\) of the centrifugal forces of the separate element of the body considered as arising from their motion of rotation about the instantaneous axis. Show further that the moment of the same centrifugal forces about the axis of resultant angular momentum is zero.
A particle of mass \(m\) at the point \((x,y)\) is acted on by a force whose rectangular components are \(X,Y\). It is found experimentally that the equations of motion of the particle in Cartesian co-ordinates are \[ m\frac{d}{dt}(\beta\dot x) = X, \quad m\frac{d}{dt}(\beta\dot y)=Y, \] where \[ \beta = \left(1-\frac{\dot x^2+\dot y^2}{c^2}\right)^{-1/2} \] and \(c\) is a constant. Show that the equations of motion in polar co-ordinates can be written in the form \[ mc^2\frac{d\beta}{dt} = P\dot r + Qr\dot\theta, \quad m\frac{d}{dt}(r^2\dot\theta\beta)=Qr, \] where \(P,Q\) are the components of force along and perpendicular to the radius vector respectively. Show that if \(P=-m\mu r^{-2}, Q=0\), the polar equation of the orbits is of the form \[ \frac{1}{r} = A+B\cos(\gamma\theta-C), \] where \[ \gamma = \left(1-\frac{\mu^2}{c^2h^2}\right)^{1/2}, \] \(h\) is the constant value of \(r^2\dot\theta\beta\) and \(A,B,C\) are constants depending on the initial conditions.
Obtain the conditions which must be satisfied by the electric intensity and the electric displacement at the interface between two dielectrics. What modifications are necessary if there is a surface charge located at the interface? The distance between the plates \(A_1, A_2\) of a parallel plate condenser is \(a_1+a_2\), and the space between them is entirely filled with two slabs of dielectric \(S_1\) and \(S_2\), of thicknesses \(a_1\) and \(a_2\), whose sides are parallel to the faces. The dielectric constants are \(K_1\) and \(K_2\). The slabs are slightly conducting, and have specific resistances \(r_1\) and \(r_2\). At the instant \(t=0\), the plate \(A_1\) (in contact with \(S_1\)) is connected to the positive pole of a battery of electromotive force \(V\), and the plate \(A_2\) is simultaneously connected to the negative pole. Prove that a charge accumulates at the interface between \(S_1\) and \(S_2\), and that at time \(t\) the surface density at the interface is \[ \frac{V}{4\pi}\frac{K_2r_2-K_1r_1}{a_1r_1+a_2r_2}(1-e^{-\alpha t}), \] where \[ \alpha = \frac{4\pi(a_1r_1+a_2r_2)}{r_1r_2(a_1K_2+a_2K_1)}. \] (The internal resistance of the battery and connecting wires and the effects of electro-magnetic induction are to be neglected.)
The magnetic vector-potential \(\mathbf{U}\) in a magnetic field \(\mathbf{H}\) is defined to be any vector function satisfying the relation \[ \mathbf{H}=\text{curl }\mathbf{U}. \] Show that the field at a point distant \(r\) from a doublet at \((x,y,z)\) of strength represented by the vector \(\boldsymbol{\mu}\) may be derived from a vector-potential given by \[ \mathbf{U} = \boldsymbol{\mu}\wedge\text{grad}\left(\frac{1}{r}\right), \] and hence that the vector-potential due to a normally magnetised magnetic shell of uniform strength \(\phi\) may be taken to be \[ \mathbf{U} = \phi\oint\frac{1}{r}d\mathbf{s}, \] where \(d\mathbf{s}\) is an element of arc of the boundary curve of the shell and the integral is taken round the boundary curve. Deduce that the mutual potential energy of two currents of intensities \(i,i'\) in closed circuits may be expressed in the form \[ -ii'\iint\frac{1}{r}d\mathbf{s}.d\mathbf{s}', \] the integrals being taken round the circuits.
The plates of a condenser of capacity \(C\) are connected by a wire of self-induction \(N\), and the system is placed in the neighbourhood of a circuit of self-induction \(L\) containing an alternating E.M.F. \(E\cos pt\). The coefficient of mutual induction is \(M\). Write down the differential equations for determining the currents \(i_1, i_2\) in the primary and condenser circuits, and deduce that the rate at which the applied E.M.F. does work exceeds the sum of the rate of expenditure of energy in heating the wires and the rate of accumulation of energy in the condenser by the amount \[ \frac{d}{dt}(\frac{1}{2}Li_1^2+Mi_1i_2+\frac{1}{2}Ni_2^2). \] Show that the phase of the current in the condenser circuit lags behind that in the primary circuit by an amount \[ \tan^{-1}\frac{pRC}{1-p^2NC}, \] where \(R\) is the resistance in the condenser circuit.
Prove that in the irrotational motion of an incompressible fluid under no forces, the pressure \(p\) at any point \((x,y,z)\) at time \(t\) is given in terms of the velocity-potential \(\phi\) by an equation of the form \[ \frac{p}{\rho}+\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial x}\right)^2+\left(\frac{\partial\phi}{\partial y}\right)^2+\left(\frac{\partial\phi}{\partial z}\right)^2\right] - \frac{\partial\phi}{\partial t} = \text{const.}, \] where \(\rho\) is the density. A circular cylinder of radius \(b\) filled with incompressible fluid at rest is suddenly given a velocity \(\omega a\) in a direction perpendicular to its axis \(A\), and is then constrained to continue moving with angular velocity \(\omega\) about a line \(O\) parallel to \(A\) and distant from it a length \(a\). Find the velocity-potential at any time \(t\), and show that the pressure at any point \(P\) in the surface of the cylinder is given by the equation \[ p-p_0 = \rho\omega^2ab\cos\theta, \] where \(p_0\) is the pressure at the axis of the cylinder, and \(\theta\) is the angle between the plane containing \(P\) and \(A\) and the plane containing \(O\) and \(A\).
Incompressible liquid of density \(\rho\) occupies the space interior to a long straight tubular membrane of circular cross-section. The surface tension \(T\) of the membrane may be considered independent of its extension, and the static pressure in the liquid is such as to make the radius of the cross-section equal to \(r_0\). Waves of small amplitude are caused to be propagated along the tube, of a type in which the membrane remains a surface of revolution and its meridian section becomes a sine curve. By considering the corresponding steady motion with reference to an observer moving with velocity \(v\), or otherwise, show that the velocity of propagation \(v\) of waves of length \(\lambda\) is given approximately by the relation \[ v^2 = \frac{T}{2\rho\lambda}\left(\frac{4\pi^2 r_0}{\lambda}-\frac{1}{r_0}\right), \] provided \(\lambda \ll 2\pi r_0\). [The velocity of the fluid is to be taken as uniform over any cross-section of the tube.]
A box in the form of a cylinder of height \(b\) with its generators vertical is divided into two parts by a horizontal partition at height \(a\), (\(a
If \(U\) and \(p\) denote the energy per unit mass and the pressure of a substance, supposed expressed as functions of the temperature \(T\) and the specific volume \(v\), establish the relation \[ \frac{\partial U}{\partial v} = T\frac{\partial p}{\partial T}-p. \] A substance is such that \[ U=av+C_vT, \] where \(a\) and \(C_v\) are constants. Show that it has an equation of state of the form \[ (p+a)f(v)=T. \] Show further that the specific heat at constant pressure \(C_p\) is independent of \(T\), and that if it is also independent of \(v\) then \[ f(v) = \frac{v+k}{C_p-C_v}, \] where \(k\) is a constant.
In a spherical triangle show that \[ \sin(B+C) = \frac{\sin A(\cos b+\cos c)}{1+\cos a}. \] The Foucault siderostat consists of a mirror which is constrained to move so that a ray from some given star \(A\) is reflected due South at all hour-angles. If \(A'\) is the point (due South) where the reflected beam from \(A\) meets the celestial sphere, and \(B'\) is the corresponding point for the ray reflected by the same mirror from some other star \(B\), prove that as the stars change in hour-angle the locus of \(B'\) is in general a small circle round \(A'\). Further, if \(\phi\) is the position-angle of \(B\) with respect to \(A\), show that when the hour-angle of \(A\) is positive (i.e. after culminating) the position-angle of \(B'\) with respect to \(A'\) is \(\phi'\), given by \[ \phi' = \phi + P\hat{A}A' + P\hat{A'}A, \] \(P\) denoting the North Pole. Hence, by means of the result quoted above for a spherical triangle, show that the rotation of \(B'\) with respect to \(A'\) is counter-clockwise or clockwise according as \(A\) is or is not circum-polar; and that in the critical case in which \(A\) just sets, the orientation of \(B'\) with respect to \(A'\) remains fixed (\(\phi'\) is measured positive in a clockwise direction).