Moving in a closed elliptical orbit, Halley's comet goes round the Sun once every 75 years. At its point of closest approach its distance from the Sun is 8.9 \times 10^7\text{ km}. If mass of the Sun is 2 \times 10^{30}\text{ kg}, what will be the greatest distance of the comet from the Sun ?
A particle is moving under a central force field. Obtain the equations of motion and prove that in such a field the areal velocity of the particle remains constant.
State Bernoulli's theorem. A horizontal pipe has a cross-section of 10\text{ cm}^2 in one region and 5\text{ cm}^2 in another. The pressure of water in the two regions is 2.370 \times 10^5\text{ N/m}^2 and 1.995 \times 10^5\text{ N/m}^2 respectively. Calculate the amount of water flowing through the pipe per minute.
The coordinates of an event in an inertial frame S are (25\text{ m}, 0, 0, 5 \times 10^{-8}\text{ s}). What will be the coordinates of this event in another interntial frame S' moving with a velocity 0.6 c in + x- direction with respect to S. The origins of the two frames were coincident at t = t' = 0
A particle of mass m_1 moving with a velocity v_1 undergoes an elastic collision with a particle of mass m_2 at rest, in laboratory-frame. After the collision the first particle moves at a certain angle to the direction of its initial velocity, and this angle is \theta in laboratory-frame and \phi in centre of mass-frame. If the ratio of masses \left(\frac{m_2}{m_1}\right) is a, show that \theta and \phi are related as \tan \phi = \left(\frac{\sin \phi}{\cos \phi + \frac{1}{A}}\right).
Define Coriolis force. Obtain an expression for the Coriolis force. How does it account for the whirling of winds in opposite directions in Northern and Southern hemispheres?
A Particle of mass m, moving with an initial velocity v_0 is acted on by a central repulsive inverse-square force, F = \frac{k}{r^2}. Show that the scattering angle \phi depends on the impact parameter b as \cot \left(\frac{\phi}{2}\right) = \frac{m v_0^2}{k} b.
Write a short note on Mass-energy equivalence.
A satellite has apogee and perigee heights above the sumace of earth as 4000 km and 640 km respectively. Calculate the eccentricity of the orbit and semi-major axis. Radius of the earth is 6400 km.
A spherical body of mass 'm' and radius 'a' is falling through a viscous fluid. If it started from rest and \eta is coefficient of viscosity of the fluid obtain an expression for its velocity as function of time. What is meant by relaxation time ? On what factors will it depend in this case?
A satellite is launched into a circular orbit close to the surface of the earth. Fine an expression for the additional velocity that has to be imparted to the satellite to overcome the gravitational pull.
What are precession an nutation ? Show that the angular velocity of precession \vec{\Omega} is related to the angular momentum \vec{L} and external torque \vec{r} as \vec{\Omega} \times \vec{L} = \vec{r} Assume \Omega \ll \omega.
Write a short note on Gyroscope.
Distinguish between stream-line motion and turbulent motion. Hence define Reynold’s number. A sphere of radius 0.1\text{ cm} moves through glycerine. A laminar flow is observed when the velocity of the sphere does not exceed 15\text{ cm/s}. At what minimum velocity of a sphere of radius 5\text{ cm} will the flow in water become turbulent? Viscosity coefficient for glycerine \eta_{\text{gly}} = 14\text{ p} and for water \eta_{\text{wat}} = 0.011\text{ p}, and density of glycerine = 1.2\text{ gm/cc}.
A stream of liquid of density \rho and speed v is directed against a plane surface at an angle \theta with the normal to the surface. After striking the surface the liquid spreads over it. Find the pressure on the surface.
The potential energy of a particle is given by U = a \cos \alpha x Obtain the positions of stable equilibrium.
The distance between the centres of oxygen and carbon atoms in a carbon monoxide molecule is 1.2\text{ \AA}. Determine the position of the centre of mass of the molecule relative to carbon. (Assume atomic masses of carbon and oxygen as 12 and 16 respectively.)
Determine the speed of an electron which has kinetic energy 1\text{ MeV}. The energy equivalent to the rest mass of an electron is 0.51\text{ MeV}.
The density of gold in its proper frame of reference is 19.3 \times 10\text{ kg/m}^3. Determine its density is a frame of reference where its velocity is 0.9 c. c is velocity of light in free space.
In the relativistic region, compare the relative increase in velocity with the relative increase in energy of a particle.