State the fundamental postulates of Einstein's special theory of relativity. Deduce Lorentz transformation equation and discuss how this accounts for the phenomenon of length contraction.

Show that the relativistic invariance laws of conservation of momentum lead to the concepts of variation of mass with velocity and mass energy equivalence.

A bead slides on a wire in the shape of a cycloid described by the equations x = a (\theta - \sin \theta) y = a (1 + \cos \theta) \quad \text{with} \quad 0 \le \theta \le 2\pi. Find the Lagrangian and equation of motion.

Write the components of the velocity 4-vector of a particle in an inertial frame S. How does the four velocity transform to another inertial frame S' ? Obtain the Einstein's law of addition of velocities from the above transformations.

A force field is given by \vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k} Is it a conservative field ? If so, what is the scalar potential ?

What is the significance of the null result of Michelson-Morley experiment ? In their experiment Michelson and Morley set l_1 = l_2 = 11\text{ metres}. The wavelength of light used was 5.5 \times 10^{-7}\text{ m}. The orbital velocity of the earth is taken to be 30\text{ kms/sec}. Estimate the fringe shift to be expected.

What are cyclic coordinates for a system of particles ? Discuss the significance of the Hamilton's principle. Show how Lagrange's equations can be derived from Hamilton's principle.

Using Euler's equations for a force free motion of a rigid body, show that the Kinetic energy remains constant throughout the motion of the rigid body.

The Lagrangian for a particle has the form L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - U(r) Obtain its form in plane polar coordinates and then get expression for the Hamiltonian using appropriate transformation.

Find the velocity and momentum of a particle having rest mass m_0 and kinetic energy equal to two times of its rest mass energy.

Discuss that the motion of a particle in a central field gets restricted to a plane provided its angular momentum is nonzero. What will happen if the angular momentum is zero?

Solve Euler's equation of motion for a rigid body to show that the torque-free motion of a spherical top is a uniform rotation about an axis fixed in space.

A particle of mass m is moving with the velocity \vec{v} on the surface of the earth. Discuss the nature of its motion in both the hemispheres due to the Coriolis force \vec{F} = -2\,m\,\vec{\omega} \times \vec{v}.

Describe briefly the Michelson-Morley experiment and discuss the significance of its null result in the context of the special theory of relativity.

Derive the law of addition of relativistic velocities. Use it to prove that under the Lorentz transformation no two velocities can add upto more than the value of the speed of light.

State Hamilton's principle for a system of particles. If I and L represent the action integral and the Lagrangian function, respectively, write down the mathematical form of Hamilton's principle and explain clearly the significance of the same.

Using the concept of D'Alembert's principle, show that the generalized force can be defined as Q_j = \sum_i \bar{F}_i \cdot \frac{\partial \bar{r}_i}{\partial q_j} , where \bar{r}_i is the Cartesian coordinate of the i^{\text{th}} particle experiencing external force \bar{F}_i and q_j stands for the generalized coordinate. Discuss the significance of the above expression.

Deduce the minimum energy of a gamma ray photon (in MeV), which can cause electron-positron pair production.

A particle of rest mass M moving at a velocity u collides with a stationary particle of rest mass m. If the particles stick together, show that the speed of the composite ball is equal to u\alpha M/(\alpha M + m), where \alpha = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} .