m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; A harmonic oscillator is represented by the equation m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; where m = 0\cdot 25\text{ kg}, \gamma = 0\cdot 07\text{ kg s}^{-1} and k = 85\text{ Nm}^{-1}. Determine (i) the period of oscillation, and (ii) the number of oscillations in which its amplitude will become half of its original value.
A light beam of wavelength 600\text{ nm} produced by a 20\text{ mW} laser source is incident on a plane mirror. Determine : (i) number of photons per second striking the surface of the mirror. (ii) force exerted by the light beam on the mirror.
Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for the threshold population inversion required for the oscillations of laser.
In a double slit Fraunhofer diffraction experiment, the slit width is 0\cdot 12\text{ mm} and the spacing between the two slits is 0\cdot 48\text{ mm}. The distance of the screen from the slits is 1\cdot 5\text{ m}. If the wavelength of the light used is 600\text{ nm}, determine (i) the missing orders of the interference maxima, and (ii) the distance between the central maxima and the first minima.
For a \text{He}-\text{Ne} laser system, what will be the magnitude of \Delta\omega_D which represents FWHM of the line shape function g(\omega), if resonant frequency \omega_0 = 3 \times 10^{15}\text{ s}^{-1} and temperature T = 300\text{ K}?
Consider a thick lens of thickness t made of a material of relative refractive index n. Let R_1 and R_2 be the radii of curvature of its two surfaces. Obtain the system matrix of the lens.
Consider multiple reflections from a plane parallel film of thickness h and refractive index n_2 and derive an expression for the total reflectivity from the surface of the film.
A cube of mass M and side ‘a’ is rotating with angular velocity \omega around one of its edges, which is, say, along the x-axis. Obtain the expressions for its angular momentum and kinetic energy. (Given that the I_{XX} = \frac{2}{3} Ma^2, I_{YX} = -\frac{1}{4} Ma^2 and I_{ZX} = -\frac{1}{4} Ma^2)
A solid shaft of mass M, length l and radius r is to be replaced by a lighter hollow shaft of the same length l and having the same ratings of \tau/\theta, where \tau is the couple and \theta is the angle of twist. Estimate the percentage reduction in mass of the hollow shaft if the outer radius of the shaft is twice the inner radius. Assume the material of the new shaft is same as that of the replaced shaft.
Consider a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger cylinder. Determine the equation of motion of the smaller cylinder.
Derive the expression for the gravitational self-energy of a uniform solid sphere of mass M and radius R.
A particle of rest mass 1\text{ kg} and velocity of magnitude 0\cdot 9c collides with a particle of mass 2\text{ kg} at rest. After collision the two particles coalesce and form a single particle of mass M and velocity V. Determine M and V.
A body moves about a point ‘O’ under no force, the principal moments of inertia at ‘O’ being 3A, 5A and 6A. The components of the initial angular velocity about the principal axes are \omega_1 = n, \omega_2 = 0 and \omega_3 = n. Find the components \omega_1, \omega_2 and \omega_3 for large values of time t.
An RC-coupled transistor amplifier has mid-frequency gain A_{\text{vm}} = 100. The values of the lower and upper cut-off frequencies are f_1 = 20\text{ Hz} and f_2 = 50\text{ kHz}. Find the frequencies at which the gain is reduced to 80.
How does supercritical magnetic field depend on temperature ? For a superconducting specimen, the critical magnetic fields are respectively 1\cdot 45 \times 10^5\text{ A/m} and 4\cdot 2 \times 10^5\text{ A/m} for 14\text{ K} and 13\text{ K}. Determine the superconducting transition temperature and the critical field at 0\text{ K}.
Give logic diagram, Boolean equation and truth tables of AND, OR and NOT gates. Show that NAND and NOR gates can be utilized as a Universal Gate.
Compare the working of FET and MOSFET with their structure, I-V curve and transfer characteristics.
How is digital system different from analog system ? What are the principal reasons for the widespread use of digital electronics ?
(i) What is the concept of effective mass of the electron ? (ii) The energy near a valence band edge is given by E(k) = -1 \times 10^{-26} k^2\text{ ergs}. An electron is removed from the orbital k = 1 \times 10^7 k_x\text{ cm}^{-1}. Find the sign and magnitude of effective mass of the hole.
The spacing between successive (100) planes in sodium chloride is 1\cdot 41\text{ \AA}. X-rays incident on the surface of the crystal are found to give rise to second order Bragg reflections at a glancing angle 10^\circ. Calculate the wavelength of X-ray radiations.
