Show that the Bulk modulus K, Young's modu- lus Y and Poisson's ratio $ \sigma $ are connected by the relation K = \frac{Y}{3(1 - 2\sigma)}.

What do you understand by streamline motion and critical velocity of a viscous liquid through a capillary tube. Capillaries of lengths $ l $, $ 2l $ and $ \frac{l}{2} $ are connected in series. Their radii are $ r $, $ \frac{r}{2} $ and $ \frac{r}{3} $ respectively. If the streamline flow is maintained and the pressure across the first capillary is $ P_1 $, deduce the pressures across the second and the third capillaries.

Derive an expression for the moment of inertia of a rigid body about any axis. What is an "ellipsoid of inertia" ? Explain clearly what you mean by the terms "principal axes" and "principal moments of inertia" ? Find the moment of inertia of a thin rectangular lamin a about an axis passing through the centre of the lamina and perpendicular to its plane. Hence determine the moments of inertia about axes passing through the midpoints of its both sides and perpendicular to its plane.

Derive an expression for the radial and transverse components of the acceleration of a particle moving in a plane. What inferences would you draw regarding the angular momentum if the transverse acceleration is zero ?

Show, using the above relation, that the equation of motion of the particle in S gets modified in S' giving rise to various fictitious forces. Identify the coriolis force and describe its effect on the flow of rivers.

Derive the relation \vec{v}_0 = \vec{v} + \vec{\omega} \times \vec{r}, where \vec{v}_0 is the velocity of a particle located at \vec{r} in a fixed frame of reference S and \vec{v} that as observed in a frame S' rotating with angular velocity \vec{\omega} with respect to S but having the common origin.

State the postulates of the special theory of relativity and based on these obtain Lorentz as well as inverse Lorentz transformations. Hence, obtain an expression to conclude that a moving clock runs more slowly than a stationary clock.

Write down the Lagrangian of a free particle in rectangular cartesian coordinates. Identify all the cyclic coordinates. Show that the constants of motion obtained from the equations for the considered form of the Lagrangian are not exactly the same as those which follow from the concept of a free particle.

The angular momentum \vec{M} of a rigid body comprising of N particles and rotating with angular velocity \vec{\omega} is given by : \vec{M} = \sum_{k=1}^{N} m_k \vec{r}_k \times (\vec{\omega} \times \vec{r}_k) where the origin coincides with the centre of mass. Express the components of \vec{M} in terms of components of the inertia tensor. Hence, show that the most general free rotation of a spherical top is a uniform rotation about an axis fixed in space.

A body of rest mass m_0 is moving in the positive direction at a velocity of 0.6\text{ C} relative to the laboratory frame. Calculate the components of the four dimensional momentum vector in the laboratory frame and in the frame of an observer who is travelling in the positive x-direction at a speed of 0.8\text{ C} relative to the laboratory frame.

An unstable particle has a lifetime of 5\text{ }\mu\text{s} in its own frame of reference and is moving towards the earth at a speed of 0.8\text{ C}. What will be the lifetime of the particle to an observer on the earth ?

Derive an equation of motion for a variable mass. Explain how it is applied in the motion of a rocket.

What is Hamilton's principle ? Obtain Lagrange's equations of motion with its help for a conservative system.

The source S' moves along the x'-axis at a speed v, and emits light at an angle \theta' to the x'-axis of its own frame. In the S-frame the emitting angle with the x-axis is \theta. Here x and x'-axes are coincident. Show that the exact relativistic aberration formula \tan \theta = \frac{\sin \theta' \sqrt{1 - v^2 / c^2}}{\cos \theta' + v} can be derived from the velocity transformation relations.

Prove that two successive Lorentz transformations are equivalent to another Lorentz transformation. Hence write down the Einstein's velocity addition relation.

Derive Euler's equations of motion for a rigid body rotating about a fixed point under the action of a torque. When a rigid body is not subjected to any net torque, write down Euler's equations of motion of the body with one point fixed.

What is centre of mass ? Show that the total linear momentum of a system of particles about the centre of mass is zero.

(a) How are linear and angular momenta related to each other ? Considering a system of mass points under the influence of forces derived from potentials only, prove that the generalised linear momentum is conserved. Establish further that the angular momentum of the above system is also conserved if the potential is centrally symmetric.

(b) Show that the mass-energy relationship in relativistic kinematics can lead to the equation E^2 = c^2 p^2 + m_0^2 c^4 where E is the total energy of the particle of rest mass m_0, linear momentum p and moving with a velocity v; c is the velocity of light in free space.

(b) What are constraints of motion ? Explain with examples the holonomic and non-holonomic constraints. Discuss critically how can one overcome the limits of constraints by introducing generalised coordinates.