A satellite, revolving in a circular equatorial orbit at height 1.36 \times 10^4\text{ km} from earth's surface from west to east, appears over some spot at the equator every 6\text{ hrs}. Show that the data are consistent with known earth's radius 6.4 \times 10^3\text{ km} and g = 9.8\text{ ms}^{-2} at earth's surface.

Write a short note on Gyroscope and its application.

Write a short note on Inertial forces in a rotating frame.

State Bernoulli's theorem. Find out the velocity of efflux of a liquid from a reservoir in which the pressure is 3.5 \times 10^5\text{ N/m}^2 above the atmospheric pressure. The density of the liquid is 700\text{ kg/m}^3.

State Kepler's laws of planetary motion. Why does a missile need an escape velocity to escape from earth?

Calculate the speed of a satellite orbiting the planet Jupiter at a distance of 108\text{ km}, above its surface (The mass and radius of Jupiter are 1.9 \times 10^{27}\text{ kg} and 69892\text{ km} respectively.)

Using Rutherford's observation that the number of \alpha particles scattered at an angle \phi and falling on unit area of the screen varied as (\text{cosec }(\phi/2))^4, deduce an expression for the probability of scattering between angles \phi and \phi + d\phi.

What do you mean by centre of mass of a system of particles? Derive expressions for the instantaneous position vector and velocity of the centre of mass of such a system of particles.

The half-life of \mu-meason at rest is 2 \times 10^{-8}\text{ sec}. Determine the half life of \mu-meason while travelling with half the speed of light in vacuum.

Write a short note on The Lorentz transformation.

Write down the expression for the dependence of mass of a particle on its velocity in special relativity. What will be the speed of a particle if its mass becomes double of its rest mass ?

Define differential scattering cross-section for a scattering process, the differential scattering cross-section for neutrons scattered elastically from a solid is of the form \alpha(\theta) = A c^{-B(\vec{K}_i \cdot \vec{K}_f)^2} where A and B are constants and \vec{K}_i and \vec{K}_f are respectively the wave vectors of the incident and scattered neutron. Determine the total scattering cross-section. [\vec{K}_i] = [\vec{K}_f]

An electron with velocity 4 \times 10^6\text{ m/s} approaches a point nucleus from a great distance with an impact parameter 0.5 \times 10^{-10}\text{ m}. Calculate the angular momentum of the electron about the nucleus. Show that the electron scattering is S-wave scattering.

Write a note on Coriolis force.

Write a note on Addition theorem of velocities in special theory of relativity.

Prove that if E and E' are respectively the neutron energies in the laboratory system, before and after collision with a nucleus of mass number A. then \frac{E'}{E} = \frac{1 + A^2 + 2 A \mu_e}{(1 + A)^2} where \mu_e is the cosine of the scattering angle in the centre of mass system.

Deduce the magnitude and direction of the acceleration of the moon and new moon.

A sphere of radius a_1 made of a material of density p_1 falls through a fluid with terminal velocity v_1. Another sphere of radius a_2 made of a material of density p_2 falls through the same liquid with terminal velocity v_2. Assuming that the viscous dreg for these particles is given by \pi \eta a v. show that the viscosity of the fluid is given by \frac{2 g a_1^2 a_2^2 (p_2 - p_1)}{9 (a_1^2 v_2 - a_2^2 v_1)}

A body falls from a great height towards the surface of the moon. Write down its equation of motion and solve it to determine the speed with which the body strikes the surface. Also find its numerical value.

Write a note on Experimental verification of variation of mass with velocity.