A transparent film of glass of refractive index 1\cdot 5 is introduced normally in the path of one of the interfering beams of Michelson interferometer which is illuminated with light of wavelength 5893\text{ \AA}. This causes 500 dark fringes to shift across the field. Determine the thickness of the film.
A rigid body is rotating under the influence of an external torque (N) acting on it. If T is the kinetic energy and \omega is the angular velocity, show that \dfrac{dT}{dt} = N \cdot \omega in the principal axes system.
A homogeneous right triangular pyramid with the base side a and height \frac{3a}{2} is shown below. Obtain the moment of inertia tensor of the pyramid:
A light rod of length 100\,\mathrm{cm} is suspended from the ceiling, horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of steel and its cross-section is 0.05\,\mathrm{sq.\ cm} and the other is of brass of cross-section 0.1\,\mathrm{sq.\ cm}. Find the position along the rod at which a weight may be hung to produce
(i) Equal stresses in both the wires,
(ii) Equal strain in both the wires. Youngs modulus of elasticity of brass and steel are 1.0 \times 10^{11}\,\mathrm{N/m^2} and 2.0 \times 10^{11}\,\mathrm{N/m^2} respectively.
(i) Calculate the mass and momentum of a proton of rest mass 1.67\times10^{-27}\,\mathrm{kg} moving with a velocity of 0.8c, where c is the velocity of light. If it collides and sticks to a stationary nucleus of mass 5\cdot0\times10^{-26}\,\mathrm{kg}, find the velocity of the resultant particle.
(ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling.
A pion at rest decays into a muon and a neutrino.
(i) Find the energy of the outgoing muon in terms of the two masses m_\pi and m_\mu (assuming m_\nu = 0). (ii) Find the velocity of muon.
On a space-time diagram after the Lorentz transformation with given v we get new tilted axis ct' and x'. For a point on the ct' axis which is at the (geometrical) distance \delta on the diagram measured from the origin, find the values of t' and t.
An observer detects two explosions, one that occurs near him at a certain time and another that occurs 2\,\mathrm{ms} later 100\,\mathrm{km} away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer?
A block of mass m is sliding on a wedge of mass M as shown in the figure below. The wedge can slide on the horizontal table. Find the equation of motion.
During take-off, an aircraft accelerates horizontally in a straight line at a rate A. A small bob of mass m is suspended on a string attached to the roof of the cabin, and a hydrogen balloon (total mass m) is tethered to the floor by a string. For each, determine the tension in the string and the equilibrium angle \theta between string and vertical. Draw a neat diagram to explain your answer.
A body of mass m at rest splits into two masses m_1 and m_2 by an explosion. After the split the bodies move with a total kinetic energy T in opposite direction. Show that their relative speed is \sqrt{\frac{2Tm}{m_1m_2}}.
(i) Define the angular velocity \omega of a rigid body rotating about some axis.
(ii) Then derive the relation \vec{v} = \vec{\omega} \times \vec{r} for the velocity of a point in the body with position vector \vec{r} relative to an origin on the axis. Draw a diagram explaining all the vectors.
Obtain the Lorentz transformations for the components of momentum-energy four vector.
A comet of mass m moves towards the Sun with initial velocity v_0. The mass of the sun is M and its radius is R. Find the total cross-section \sigma for striking the Sun. Take the Sun to be at rest and ignore all other bodies.
An electron is moving under the influence of a point nucleus of atomic number Z. Show that the orbit of the electron is an ellipse.
The radius of the Earth is 6.4 \times 10^{6}\ \mathrm{m}, its mean density is 5.5 \times 10^{3}\ \mathrm{kg/m^{3}} and the universal gravitational constant is 6.66 \times 10^{-1}\ \mathrm{Nm^{2}/kg^{2}}. Calculate the gravitational potential on the surface of the Earth.
A particle moving in a central force field describes the path r=ke^{\alpha\theta}, where k and \alpha are constants. If the mass of the particle is m, find the law of force.
A capillary tube having 1.0\,\mathrm{mm} diameter, 20\,\mathrm{cm} in length is fitted horizontally to a vessel in which alcohol is kept fully up to the neck. Density of alcohol is 8 \times 10^{2}\,\mathrm{kg/m^3}. The depth of the centre of the capillary tube below the surface of alcohol is 40\,\mathrm{cm}. Find the amount of alcohol that will flow out of the capillary tube in 10 minutes. Coefficient of viscosity of alcohol is 0.0012\,\mathrm{N\,s/m^2}.
What is Josephson Effect ? Give an account of applications of superconductivity.
Consider an amplifier with an open-loop (no feedback) gain of A and a feedback factor \beta. Derive the expression for the gain with feedback, A_f. Derive the condition for the amplifier with feedback to act as an oscillator. Comment on the change in A_f with a change in A.