A uniform rod \(AB\), of mass \(m\) and length \(a\), is free to turn about a fixed point \(A\). The end \(B\) is connected by an elastic string, of natural length \(l_0\) and modulus \(\lambda\), to a point \(C\) distant \(d\) from \(A\) and vertically above it. Show that the steady motions, in which the rod rotates with angular velocity \(\omega\) about a vertical axis and makes a fixed angle \(\alpha\) with the vertical, are stable. Show that to any given value of \(\alpha\) there is in general one and only one value of \(\omega\), but that if \[ \lambda = \frac{mgl_0(a^2+d^2)^{3/2}}{2d(a^2+d^2)^{3/2}-l_0^3} \] the steady motion \(\alpha=\pi/2\) is possible with any value of \(\omega\); and show that in this case the period of a small oscillation about the steady state is \[ 2\pi\left[\omega^2 + \frac{3\lambda d^2}{m(a^2+d^2)^2}\right]^{-1/2}. \]
State the principle of virtual work as applied to impulses. Four heavy uniform rods, smoothly jointed together at their ends, rest on a smooth horizontal table in the form of a square \(ABCD\), and the joints \(A, C\) are connected by a light inextensible strut \(AC\). A blow of impulse \(I\) is applied at \(A\), in the direction and sense of \(AB\). Show that the impulsive thrust in the strut is \(I/\sqrt{2}\).
If \(\mathbf{A, B, C}\) are three linearly independent vectors, show that necessary and sufficient conditions for a vector \(\mathbf{P}\) to be equal to a vector \(\mathbf{Q}\) are \[ \mathbf{P.A=Q.A, P.B=Q.B, P.C=Q.C}. \] A point \(O\) of a rigid body is fixed. If \(\mathbf{v_1, v_2}\) are vectors representing the velocities of any two points \(K_1, K_2\), show that \[ \mathbf{v_1.r_2+v_2.r_1=0}, \] where \(\mathbf{r_1, r_2}\) are the vectors \(\mathbf{OK_1, OK_2}\). Show further that, provided \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are not parallel, the equations \[ \mathbf{\Omega\wedge r_1 = v_1, \quad \Omega.v_2 = 0} \] define a unique vector \(\mathbf{\Omega}\) which has the property that \[ \mathbf{\Omega\wedge r_2 = v_2}, \] and that if \(\mathbf{v_r, r_r}\) refer to any other point \(K_r\) of the rigid body then \[ \mathbf{\Omega\wedge r_r=v_r}. \] [The notation \(\mathbf{X.Y}\) denotes a scalar product, \(\mathbf{X\wedge Y}\) a vector product.]
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.]