A system with 2 independent coordinates x_1 and x_2 has the following Larangian L = \frac{1}{2} \frac{(\dot{x}_1)^2}{\alpha + \beta x_2} + \frac{1}{2} (\dot{x}_2)^2 (x_2)^2 - \gamma x_2 \alpha, \beta, \gamma being constants. Obtain the Lagrangian equation of motion.

Define scattering cross-section. A charged particle of mass m and charge Z_e is scattered by another charged particle of charge Z_e at rest. Deduce the expression for the scattering cross-section.

A star of the size of our sun having a radius of 7 \times 10^5\text{ km} and having a mass 2 times as that of the sun is rotating at a speed of one revolution every 10 days. If the star undergoes a gravitational collapse to a neutron star of radius 10\text{ kms}, assuming that the star is a uniform sphere at all times, calculate the rotation speed of the neutron star. The moment of inertia of solid about its diameter is 2/5\text{ MR}^2 where M and R are the mass and radius of the solid sphere respectively.

Obtain the equation of motion of a particle moving relative to a rotating frame of reference. Explain the term representing coriolis force in this expression.

Neptune's period of revolution is 165\text{ yr}. What is the average radius of Neptune's orbit in teams of the Sun-Earth distance?

A top is spinning about its axis of symmetry. If its spin angular momentum is large compared to the change caused by the applied torque the top processes. If it is not spinning, the top would tip over. Show what happens in between, when the top spins slowly. Derive an approximate expression for the angular speed of the precessional motion.

An aeroplane makes a complete half circle of 50\text{ m} radius towards left when flying at 200\text{ km / hour}. The rotary engine and the propeller of the plane weigh 400\text{ kg} having a radius of gyration of 30\text{ cm}. The engine rotates at 2400\text{ rpm} clockwise viewing from the rear. Find the gyroscopic couple on the aircraft.

Show that the combined effects of translation of the centre of mass and rotation about an axis through the centre of mass are equivalent to a pure rotation with the same angular speed about an axis through the point of contact of a rolling body.

Ram and Shyam are two skaters weighing 40\text{kg} and 60\text{ kg} respectively. Ram travelling at 4\text{ m/s} meets Shyam travelling at 2\text{ m/s} in opposite direction and collides head-on.

(i) If they remain in contact what is their final velocity ?

(ii) How much kinetic energy is lost ?

The mass of the Moon is about 0.13 times the mass of the Earth. The distance from the centre of the Moon to the centre of the Earth is 60 times the radius of the Earth. Taking the Earths radius to be 6378\text{ km}. find out the distance of the centre of mass of the Earth - Moon system from the centre of the Earth.

Derive an expression for the mass-energy equivalence using the principle of special relativity.

What is 'Loren & Contraction' ? A spaceship of rest length 100\text{ mm} takes 4\ \mu\text{s} to pass an observer on Earth. What is its velocity relative to the Earth?

Write a short note on Coriolis force.

Using Bernoulli's principle indicate how to compute the thrust on a rocket produced by the escape of its exhaust gases.

Starting from the definition of angular momentum of a single particle, obtain an expression for the angular momentum of a rotating rigid body. Hence, discuss the time derivative of the angular momentum and deduce the law of conservation of momentum.

Distinguish between elastic and inelastic collision. Find the minimum distance an \alpha-particle with kinetic energy of 0.4 Mev can approach a stationary but free Lithium nucleus in a head-on collision.

Write a short note on Applications of Bernoulli's equation.

Discuss the Reynolds number for describing the stability of fluid flow, Air streams horizontally across an aeroplane wing of area 4\text{ m}^2 and weighing 300 kg. The air speed is 60 m/s and 40 m/s over the top surface and under the bottom surface respectively. Find (i) the lift on the wing and (ii) the net force on it, given that the density of air is equal to 1.293\text{ kg/m}^3.

Consider the motion of a rocket in a gravitational field and derive an expression for its final velocity when the fuel burns at a constant rate till it is fully consumed.

Explain the precession of a spinning top and show that precssional angular velocity is independent of the angle of inclination. A top is spinning with 20 rev/s about an axis inclined at 30^\circ with the vertical. Its radius of gyration is 5 cm. The centre of mass is \pi cm from the pivot point. Calculate the frequency of precession.