PYQ Bank

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.

A meson of rest mass $ \pi $ comes to rest and disintegrates into a muon of rest mass $ \mu $ and a neutrino of zero rest mass. Show that the kinetic energy of motion of the muon is T = \frac{(\pi - \mu)^2 c^2}{2\pi} .

The length of a moving rod can be defined as the product of its velocity and the time interval between the instants that both the end points of the rod pass a fixed mark in S system. Show that this definition leads to the space contraction.

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.

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.

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 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.

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.

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.

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.

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

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.

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

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

(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.

3. (a) Show that the length L of an object moving with a velocity v is given in the direction of motion by L = L_0 \left(1 - \frac{v^2}{c^2}\right)^{1/2} where L_0 is the proper length and c is the velocity of light in free space. What will be the shape of a spherical ball while moving under relativistic regime ?

2. (a) Considering the scattering of \alpha-particles by the atomic nuclei, find out the Rutherford scattering cross-section. Explain the physical significance of the final expression.