Explain the meaning of partial differentiation. If \(f(x,y)=0\) and \(\phi(y,z)=0\), shew that \[ \frac{\partial f}{\partial y}\frac{\partial\phi}{\partial z}\frac{dz}{dx} = \frac{\partial f}{\partial x}\frac{\partial\phi}{\partial y}. \]
Obtain the usual differential equation \(EI\frac{d^4y}{dx^4}=w\) for the deflection of a uniform heavy beam when the deflection is everywhere small. A uniform beam of length \(l\) is supported at the ends; prove that the greatest droop occurs at the middle point and is \[ \frac{5wl^4}{384EI}. \]
An engine is pulling a train and works at constant power H. If M is the mass of the whole train and F the (constant) resistance, prove that the time of generating a velocity \(v\) from rest is
\[ \frac{MH}{F^2}\log\frac{H}{H-Fv} - \frac{Mv}{F} \]
seconds, provided \(v
Prove that a circular orbit described under a central force varying as \(r^{-s}\) is stable if and only if \(s<3\). Obtain an expression for the apsidal angle and show that it can only equal \(\pi\) if \(s=2\). Find approximately the progressive motion of the apse when \(s\) differs from 2 by a small constant amount.
A smooth non-circular disc is rotating with angular velocity \(\omega\) on a smooth horizontal plane about its centre of mass, when it strikes a smooth uniform rod of mass \(m\) at the middle point of the rod. Prove that the new angular velocity is \[ \frac{(M+m)I-ep^2Mm}{(M+m)I+p^2Mm}\omega, \] where M and I are the mass and moment of inertia of the disc, p the perpendicular from its centre of mass to the normal at the point of contact, and e the coefficient of restitution.
Obtain the equations of motion of a symmetrical spinning top free to rotate under gravity about a fixed point in its axis of symmetry. Show that such a top is capable of executing a steady precession inclined at an angle \(\theta\) to the vertical at either of two different rates, which when the axial spin \(n\) is very large approximate to \[ \frac{Cn}{A\cos\theta}, \quad \frac{Mgh}{Cn}, \] where A and C are the transverse and axial moments of inertia through the fixed point, M is the mass of the top, and h the distance of the centre of gravity from the fixed point.
A sphere is set rolling on a horizontal plane which is made to rotate about a fixed vertical axis with constant angular velocity \(\omega\). Prove that with suitable initial conditions the path in space of the centre of the sphere is a circle described with angular velocity \[ \frac{k^2\omega}{a^2+k^2}, \] where \(a\) is the radius and \(k\) the radius of gyration of the sphere about a diameter.
Calculate the kinetic energy of a thin spherical shell of gravitating matter of mass M when it has fallen in symmetrically from infinity to radius \(r\), and so verify the expression \(M^2/2r\) for the gravitational energy. Show that for a sphere of radius \(a\) in which the density is a function of the radius only \[ \frac{1}{2}\iiint V\rho dv = \int_{r=0}^{r=a} \frac{M(r)dM(r)}{r}, \] where V is the potential at any point, M(r) is the mass of the sphere inside the sphere of radius \(r\), and the volume integral is taken over the whole sphere.
Describe shortly the part played by the polarization in the electrostatic theory of dielectrics. Show that the polarization inside a sphere whose specific inductive capacity is \(\epsilon\) (constant) and radius \(a\), placed in a uniform field of intensity F, is everywhere \[ \frac{3F}{4\pi}\frac{\epsilon-1}{\epsilon+2}, \] and that the internal potential is \(Fz\frac{3}{\epsilon+2}\).