Find expressions for the components of acceleration along and perpendicular to the radius vector of a point whose polar coordinates are known as functions of time. Find the law of force under which a particle can describe the spiral \(r=a\theta+b\), and find the velocity at any point of the path.
A uniform circular hoop of radius \(r\) rolls steadily on a horizontal plane so that its centre describes with velocity \(V\) a horizontal circle of radius \(R\). Its plane makes a constant angle \(\alpha\) with the horizontal. Prove that \[ V^2 = \frac{2gR^2\cot\alpha}{4R+r\cos\alpha}. \]
Prove that the small oscillations of a dynamical system about a position of equilibrium are compounded of a number of simple modes for each of which all coordinates of the system execute simple harmonic oscillations. Find the periods of oscillation in a vertical plane of a system consisting of two equal uniform rods \(AB, BC\) jointed together at \(B\) and hung from a joint at \(A\) so that they are vertical in their equilibrium position.
Explain the method of images in electrostatics. Two dielectrics of specific inductive capacity \(K_1\) and \(K_2\) are separated by an infinite plane face. Two charges \(e_1\) and \(e_2\) are placed at any points in the two media. Find the direction and magnitude of the force which acts on the charge \(e_1\). (The line joining \(e_1\) and \(e_2\) is not necessarily perpendicular to the interface.)
Define the coefficients of capacity \(q_{rs}\) for a system of conductors and show that \(q_{rs}=q_{sr}\). Find the coefficients of capacity for a system consisting of three concentric spheres of radii \(a,b,c\).
Prove that a solid gravitating sphere attracts external bodies as though its whole mass were concentrated at the centre. Gravitating matter is uniformly distributed between two concentric spherical surfaces of radii \(a\) and \(b\), (\(a>b\)). A particle starting from rest at the outer surface falls through a smooth hole into the hollow core, acquiring a velocity \(v\) in doing so. If \(c\) is the velocity acquired by a particle in falling from infinity to the outer surface show that \[ \left(\frac{v}{c}\right)^2 = \frac{a^2+2b^3-3ab^2}{a^3}. \]
Find the differential equation which must be satisfied by magnetic potential in a magnetic material the intensity and direction of the magnetisation of which is known. A steel sphere is magnetised so that the direction and intensity of magnetisation is constant. It is surrounded by a close fitting shell of paramagnetic material of permeability \(\mu\). The outer radius of the shell is \(n\) times the radius of the steel sphere. Show that the magnetic potential at external points is \[ \frac{9M\mu n^3}{\{2(\mu-1)^2-n^3(2\mu+1)(\mu+2)\}} \left(\frac{\cos\theta}{r^2}\right) \] where \(M\) is the total magnetic moment of the steel sphere.
A circular wire of radius \(a\) and carrying a current \(i\) is placed so that its centre is at a distance \(c\) from an infinite vertical wire carrying a current \(i'\). The plane of the circular wire is vertical and perpendicular to the vertical plane which contains the straight wire and the centre of the circular wire. Show that a couple of magnitude \[ 4\pi ii'c\left(1-\frac{c}{\sqrt{a^2+c^2}}\right) \] acts on the circular wire tending to turn it about its vertical diameter.
A telegraph cable has resistance \(r\) per unit length and electrostatic capacity \(c\) per unit length. The conductivity between the core and the surrounding sea water through the insulating covering is \(\kappa\) per unit length. Neglecting the effects of self induction find the differential equation which the current must satisfy when a variable E.M.F. is applied at one end. A periodic electromotive force of period \(\frac{2\pi}{p}\) is applied at one end, show that if the wire is infinitely long the amplitude of the electric oscillations will be reduced in the ratio \(1:e\) at a distance \(\frac{1}{2}\sqrt{\kappa r}\left[1+\left(1+\frac{c^2p^2}{\kappa^2}\right)^{\frac{1}{2}}\right]^{-\frac{1}{2}}\) from the end.
Discuss from a thermodynamical point of view the connection between the osmotic pressure of a salt solution and the lowering of the vapour pressure of the solvent due to the presence of the salt.