Prove the addition theorem of velocities in Special Theory of Relativity. Two bodies A and B are moving away in opposite directions each with a speed of 0.70\text{ c} with respect to a stationary observer. Deduce the speed or B as measured by A.
Write down Lorentz transformation relations and prove that x^2 + y^2 + z^2 - c^2 t^2 is invariant under this transformation. An event occurs at x^1 = 60\text{ m} at t^1 = 8 \times 10^{-8}\text{ s} in a reference frame S' which is moving along the common x or x' axis with a speed 3c/5 with reference to a stationary from S. The origins of the two frames coincide at t=0, t'=0. Deduce the space time coordinates of the event in the frame S.
A pipe of varying diameter is used to lift water by 7\text{ m} the area of cross-section of the base is 125\text{ cm}^2 and the pressure here is 2.5 \times 10^5\text{ N/m}^2. The area of cross-section of the top is 25\text{ cm}^2. The rate of flow of water is 3 \times 10^{-2}\text{ m}^3/\text{s}. Calculate the pressure of water at the top neglecting energy losses.
Write a note on Gyroscope and its applications.
A thin uniform rod of length l and mass m is hinged to the floor at its lower end. It begins to fall from a vertical position. Deduce an expression for the angular speed of the road when it hits the floor. How would you explain that the rod would suffer less damage if the floor is soft than if the floor is hard?
State Kepler's laws of planetary motion. Assume the law of gravitation to be of the form F = mMG/r^n where n is some number. Find the value n which will be consistent with Kepler's third law. For this you may assume planetary orbits to be circular.
A satellite moves round the earth at a distance of 3.884 \times 10^5\text{ km} from its centre. Find its period of revolution in days. Proceed dto deduce the distance for a geostationary satellite.
Define differential, scattering cross-section. Write down the dependence of Rutherford scattering cross-section \sigma(\theta), on the scattering angle \theta and sketch this dependence graphically. In the present case, the total scattering cross-section, \sigma = \int \sigma(\theta) d\Omega turns out to be infinite. Comment on this result.
A neutron of energy 1.00\text{ MeV} collides with a stationary helium nucleus and is scattered. Deduce the momentum of the neutron and the helium nucleus in their centre of mass system.
A girl on skis (mass 60kg including skis) reaches the bottom of a hill going 20\text{ ms}^{-1}. What is her momentum? She then strikes a snowdrift which slops her on 3s. What average force does the snow exert on the girl? Assuming that the force remains constant. How far does the girl penetrate the snowdrift?
State how Rutherford's experiments on alpha scattering by thin gold foils established that
(i) the positive charge in an atom is confined to a nucleus of radius far less than the atomic radii, and
(ii) the Coulomb law is valid down to such small distances. Why do the electrons not participate in the interactions appreciably?
Write a short note on Mass-Energy equivalence.
A solid cylinder released from rest rolls (without slipping) downs piano of length l inclined at angle \theta with the horizontal. Deduce an expression for the velocity of the centre of mass of the cylinder at the bottom of the plane, neglecting friction.
State Bernoulli's theorem for an ideal liquid. Deduce the related equation from first principles and describe one application in relation to sonic measurement technique or device.
The spike heel on a girl's shoe is a square, 1.0 cm on an edge. If her mass is 50kg, calculate (in atmospheres) the pressure exerted on the ground when she balances on one heel.
A spinning top shows precession. Explain how the precession rate changes as the spinning becomes slower and as the inclination of the spinning axis from the vertical increases.
Define Coriolis force and write an expression for it. Through suitable examples, explain the way this force varies in different parts of the earth's surface and for different velocities of the concerned particle.
What is the angular momentum of an earth satellite of mass 100 kg which moves in a circular orbit of radius equal to twice the radius of the earth? Derive the relation used.
The mass of a molecule of DNA from T2 bacteriophage is 1.2 \times 10^8\text{ a.m.u.} The molecule is a double helix of 1.2 \times 10^4 turns and the radius of gyration of the helix is 6.7\text{ \AA}. If the rotational energy of the molecule about the axis of the helix is equal to thermal energy at 300 K, deduce the value of \omega, the angular frequency of rotation.
Write a short note on Motion of rocket under constant force field.
