A particle of mass \text{m} is suspended from a spring which has mass \text{m} and force constant \text{k}. Show that the oscillation frequency is given by \omega = \frac{\sqrt{3}}{2}\omega_0, where \omega_0 is frequency of oscillation when spring is considered massless.

Explain the construction and working of a quarter-wave plate. How is it used to produce circularly and elliptically polarized light?

Describe the interference by a plane parallel thin film when illuminated normally and obliquely by a plane wave. Also list some important practical applications of the thin film interference phenomenon.

Determine the position of focal points, principal points and nodal points for a spherical lens of radius 20\text{ cm}. The refractive index of the material of the lens is 3/2. Indicate the positions of all the points in a diagram.

What do you understand by pulse dispersion in multimode optical fibres? Deduce an expression for intermodal pulse dispersion in a step-index fibre. Also compare the pulse dispersions of the step-index fibres with the parabolic-index fibres.

A 15\cdot 0\text{ cm} violin string, fixed at both the ends, is vibrating with the largest wavelength. The speed of the wave in this string is 250\text{ m/s}, and the speed of sound in air is 350\text{ m/s}. What is the frequency and wavelength of the emitted sound wave?

A parallel beam of sodium light of wavelength 5893\text{ \AA} is incident on a diffraction grating. The angle between the first order spectra on either side of the normal is 27^{\circ} 42'. Find :

(i) The number of rulings/mm on the grating.

(ii) The greatest number of bright images obtained.

Within the framework of the Cornu's spiral, describe the intensity distribution at an arbitrary point \text{P} when a plane wave is incident normally on a long narrow slit of width \text{b}. For this case, under what condition will the diffraction pattern be of Fraunhofer type?

Calculate the group and phase velocities for the wave packet corresponding to a relativistic particle.

Consider a damped harmonic oscillator given by the equation \ddot{x} + 2\mu \dot{x} + \omega_0^2 x = 0. Obtain the displacements for (i) overdamped, (ii) underdamped and, (iii) critically damped motion.

Draw the energy level diagram of He-Ne laser. The metastable state of ruby laser is at 1.786\text{ eV}. Calculate the wavelength of the light emitted.

(i) What are the similarities and dissimilarities between a zone plate and a convex lens ? (ii) A monochromatic light of wavelength 5860 \times 10^{-8}\text{ cm} is incident normally on a 2\text{ cm} wide grating. The first order spectrum is produced at an angle of 20^\circ with respect to the normal. Determine the total number of lines on the grating. How does the resolving power of grating depend on the number of rulings on the grating ?

What are the special features of step-index fibre ? The step-index fibre has core glass and cladding glass with refractive indices 1.68 and 1.5 respectively. Calculate the numerical aperture. Find the critical angle and acceptance angle for entrance of light if the fibre is placed in air.

(i) Distinguish between interference fringe pattern and diffraction fringe pattern. (ii) In an experiment using Newton's rings, the diameter of 5^{\text{th}} and 15^{\text{th}} rings were found to be 33.6 \times 10^{-4}\text{ m} and 59 \times 10^{-4}\text{ m} respectively. Calculate the radius of curvature of the plano-convex lens if the source of light is sodium light (\lambda = 589.0\text{ nm}). Mention any other applications of Newton's ring experiment.

Explain the phenomenon of double-refraction in calcite crystal with special reference to Ordinary Ray (OR), Extraordinary Ray (ER) and uniaxial negative crystal.

Explain with proper examples the techniques of obtaining interference by 'division of wavefront' and 'division of amplitude'.

Explain the formation of spectra by a plane diffraction grating. What particular spectra would be absent if the width of the slit is equal to the width of the opaque spacing between any two consecutive slits?

An achromatic doublet of focal length 25\text{ cm} is to be made by placing a convex lens of borosilicate crown glass in contact with a diverging lens of dense flint glass. Assuming n_r = 1 \cdot 61864, n_b = 1 \cdot 62468, n_r' = 1 \cdot 71426 and n_b' = 1 \cdot 72906, calculate the focal length of each lens. Here, the unprimed and the primed quantities refer to the borosilicate crown glass and dense flint glass respectively.

Explain an experiment for determination of the refractive index of a gas using Michelson interferometer.

Consider a thick equiconvex lens of refractive index 1 \cdot 5 as shown in the figure. The magnitude of the radii of curvatures of the two surfaces is 5\text{ cm}. The thickness of the lens is 1\text{ cm} and the lens is placed in the air. Obtain the system matrix and determine the focal length and the position of unit planes :