PYQ Bank

Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio \sigma are related by the equation Y = 2n(1 + \sigma).

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 body moving in an inverse square attractive field traverses on elliptical orbit with eccentricity e and period \gamma. Find the time taken by the body to traverse the half of the orbit that is nearer the centre of force. Explain briefly why a comet spends only 18\% of its time on the half of its orbit that is nearer the sun.

Calculate the percentage contraction in the length of a rod in a frame of reference, moving with velocity 0.8c in a direction What is the orientation of the rod in the moving frame of reference in case (ii)?

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 ?

Given a proton for which \beta=0.995 measured in the laboratory. What are the corresponding relativistic energy and momentum? Take, m_p=1\cdot67\times10^{-24}\,\mathrm{g}.

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.

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.

Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?

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.

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.

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

Using Poiseuille's formula, show that the volume of a liquid of viscosity coefficient \eta passing per second through a series of two capillary tubes of lengths l_1 and l_2 having radii r_1 and r_2 is obtained as Q=\frac{\pi p}{8\eta\left[\frac{l_1}{r_1^4}+\frac{l_2}{r_2^4}\right]}. where p is the effective pressure difference across the series.

Define coefficients of viscosity and kinematic viscosity of a fluid. What are Poise and Stokes?

Write down Poiseuille's formula and mention its limitations in analysing the flow of a liquid through a capillary tube.

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?

Show that the differential scattering cross-section can be expressed as \sigma(\theta)=\frac{s}{\sin\theta}\left|\frac{ds}{d\theta}\right|, where s is the impact parameter and \theta is the scattering angle.

Draw a neat diagram to explain the scattering of an incident beam of particles by a centre of force.

Prove mathematically that the addition of any velocity of a particle to the velocity of light in free space merely reproduces the velocity of light in free space only.