UPSC Physics Optional PYQs (CSE & IFoS)PYQs

In special theory of relativity, the Hamiltonian of a free particle with rest mass m_0 is given by \sqrt{p^2 c^2 + m_0^2 c^4}. Obtain the Lagrangian of the system using Legendre transformation.

Consider a homogeneous cube of total mass M and side a. Taking the origin at one corner of the cube and axes along the edges of the cube, construct the moment of inertia tensor. Calculate principal moment of inertia.

Show that the wave equation for the propagation of electromagnetic scalar potential \phi(x, y, z, t) \left[\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right] \phi(x, y, z, t) = 0 remains invariant under Lorentz transformation.

A particle moving in a central force field located at r=0, describes a spiral, r=e^{-\theta}. Find the force law.

(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.

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

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.

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.

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

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

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.

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}.

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?

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 ?

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.

Four solid spheres A, B, C, and D each of mass m and radius a, are placed with their centres on the four corners of square of side b as shown in the figure below:

Calculate the moment of inertia of the system about one side of the square, Also, calculate the moment of inertia of the system about a diagonal of the square.

Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio \sigma are related by the equation Y = 2n(1 + \sigma).

A horizontal pipe of non-uniform bore has water flowing through it such that the velocity of flow is 40\ \mathrm{cm/s} at a point where the pressure is 2\ \mathrm{cm} of mercury column. What is the pressure at a point where the velocity of flow is 60\ \mathrm{cm/s}? (Take, g = 980\ \mathrm{cm/s^2} and density of water = 1\ \mathrm{g/c.c.})

Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?