A student is working in a physics laboratory, which is at temperature 27^{\circ}\mathrm{C}, on a sonometer to study formation of stationary waves. The cross-sectional area of the sonometer wire is 0.85\times10^{-6}\ \mathrm{m^2} and a tension of 20\ \mathrm{N} is applied on it. If the rigid supports are 1.2\ \mathrm{m} apart and the temperature of the wire drops by 7^{\circ}\mathrm{C}, calculate the (i) final tension and (ii) fundamental frequency of vibration of the wire. Take, coefficient of linear expansion and isothermal Young's modulus as 1.5 \times 10^{-5}\ \mathrm{K}^{-1} and 2.0 \times 10^{11}\ \mathrm{N\,m}^{-2} respectively.

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)?

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.

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

How does one obtain the angular velocity of the Earth about the North Pole with respect to a fixed star as 7.292 \times 10^{-5}\,\mathrm{sec}^{-1}? Explain your method of calculating the above value.

Show that the moment of inertia of a circular disc of mass M and radius R about an axis passing through its centre and perpendicular to its plane is \frac{1}{2}MR^2.

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.

Show that the rest mass energy of an electron is 0.51\,\mathrm{MeV}. (use the standard values of the physical parameters).

Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms.

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

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.

A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time period of the motion.

Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression?

A body turns a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.

A sphere of radius R moves with velocity \vec{u} in an incompressible, non-viscous ideal fluid. Calculate the pressure distribution over the surface of the sphere. Do you think that a force is necessary to keep the sphere in uniform motion?

Consider a rigid body rotating about an axis passing through a fixed point in the body with an angular velocity \vec{\omega}. Determine the kinetic energy of such a rotating body in a coordinate system of principal axis. If the earth suddenly stops rotating, what will happen to the rotational kinetic energy? Comment in detail.

A mirror is moving through vacuum with a relativistic speed v in the x-direction. A beam of light with frequency \omega_i is normally incident (from x=\infty) on the mirror.

If I' and I be the Moments of Inertia of a body about an axis passing through an arbitrary origin and about a parallel axis through the centre of mass respectively, show that I' = MR^2 + I, where \vec{R} is the position vector of the centre of mass with respect to the arbitrary origin and M is the mass of the body.

The density inside a solid sphere of radius a is given by \rho = \frac{\rho_0 a}{r}, where \rho_0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.