PYQ Bank

State and explain Fermat's principle of extremum path and use the same to deduce the laws of reflection and refraction of light.

Obtain the system matrix for a thin lens placed in air and made of material of refractive index 1.5 having radius of curvature 50\,\mathrm{cm} each. Also find its focal length.

Find the velocity of sound in a gas in which two waves of wavelengths 1.00\,\mathrm{m} and 1.01\,\mathrm{m} produce 10 beats in 3 seconds.

A bead slides on a wire in the shape of a cycloid described by the equations x = a (\theta - \sin \theta) y = a (1 + \cos \theta) \quad \text{with} \quad 0 \le \theta \le 2\pi. Find the Lagrangian and equation of motion.

State the fundamental postulates of Einstein's special theory of relativity. Deduce Lorentz transformation equation and discuss how this accounts for the phenomenon of length contraction.

Show that the relativistic invariance laws of conservation of momentum lead to the concepts of variation of mass with velocity and mass energy equivalence.

Describe Michelson-Morley experiment and show how the negative results obtained from this experiment were interpreted.

State and explain Stokes' law. A drop of water of radius 0.01\ \mathrm{m} is falling through a medium whose density is 1.21\ \mathrm{kg/m^3} and \eta = 1.8 \times 10^{-5}\ \mathrm{N\,s/m^2}. Find the terminal velocity of the drop of water.

(i) State and prove Hamilton's principle and use it to prove that the shortest distance between two points in space is a straight line joining them. (ii) Use Hamiltonian mechanics to find the differential equation for planetary motion, moving under force f(r) = -\frac{k}{r^2} and prove that the areal velocity is constant.

Express angular momentum in terms of kinetic, potential and total energy of a satellite of mass m in a circular orbit of radius r.

Prove that x^2+y^2+z^2=c^2t^2 is invariant under Lorentz transformation.

Discuss the mechanics of a system of point particles with special emphasis on the conservation theorems. How can we extend the results to a system with continuous mass distribution ?

Define moment of inertia and explain its physical significance. Calculate the moment of inertia of an annular ring about an axis passing through its centre and perpendicular to its plane.

A ball moving with a speed of 9\ \mathrm{m/s} strikes an identical stationary ball such that after the collision the direction of each ball makes an angle 30^\circ with the original line of motion. Find the speed of the balls after the collision. Is the kinetic energy conserved in this collision?

A diatomic molecule can be considered to be made up of two masses m_1 and m_2 separated by a fixed distance r. Derive a formula for the distance of centre of mass, C, from mass m_1. Also show that the moment of inertia about an axis through C and perpendicular to r is \mu r^2, where \mu=\dfrac{m_1m_2}{m_1+m_2}.

What are logic gates and truth tables? Obtain the truth table of a two-input OR gate.

Give an elementary outline of high-temperature superconductors and their experimental status.

Describe the following laws of Boolean algebra with one application each to logic circuit :

(i) Commutative law of addition

(ii) Associative law of addition

(iii) Associative law of multiplication

(iv) Distributive law

With the help of a schematic diagram, show how entropy and specific heat vary with temperature for a superconductor.

Distinguish between diamagnetic, paramagnetic and ferromagnetic substances.