A three-particle system consists of masses \text{m}_i,\ i = 1, 2, 3 and their respective coordinates (\text{x}_i, \text{y}_i, \text{z}_i) : m_1 = 3m,\ (x_1, y_1, z_1) = (a, 0, a), m_2 = 4m,\ (x_2, y_2, z_2) = (a, a, -a), m_3 = 2m,\ (x_3, y_3, z_3) = (-a, a, 0), where m and a are positive constants. Calculate the inertia tensor of the system.

A mouse of mass m jumps on the outside edge of a freely turning ceiling fan of rotational inertia I and radius R. By what ratio does the angular velocity change ?

Where do you find the applications of gyroscope? A top of mass 0.200\ \mathrm{kg} is made up of a thin disc of radius 0.12\ \mathrm{m}. It is pierced in the centre and a pin of negligible mass is mounted normal to its plane. The pivot under the disc is 0.03\ \mathrm{m} long. The top is made to spin with its axis making an angle \theta = 20^\circ with the vertical and a precessional angular speed of 2\ \mathrm{rad/s}. Calculate the angular speed with which it spins.

Write down Euler's dynamical equations of motion (no derivation) of a rigid body about a fixed point under the action of a torque. Show that the kinetic energy of the torque-free motion is constant.

Explain drift velocity, mobility and conductivity of an intrinsic semiconductor.

Calculate the total conductivity in an intrinsic semiconductor. How does this conductivity change with temperature ?

What are operational amplifiers? How can it be used as an inductor? Prove it mathematically.

Derive the expression for electron concentration in the conduction band of an intrinsic semiconductor.

Draw the circuit of a phase shift oscillator using transistor. Obtain the expression for the frequency of oscillations and condition for sustained oscillations. Give necessary equivalent circuit while deriving the expression of frequency of oscillations and condition for sustained oscillations.

Derive Bragg diffraction law for X-ray diffraction. Compare Laue and Debye-Scherrer methods for crystal structure determination.

Estimate the current gain of an n-p-n transistor if only 0\cdot 3\% of the conduction electrons that enter the base from the emitter recombine in the base region.

Derive an expression for lattice specific heat in Debye model. Find its low temperature limit (Debye T^3 law).

A small magnet is placed on the surface of a disc of normal material. Explain what happens when the disc is cooled down so that it becomes superconducting. Can this be explained from classical viewpoint if we simply treat the superconductor as a material which has zero resistivity ?

Find the output voltage of the three-input adder circuit shown below.

Describe the working of a microprocessor system in block diagram. How is its performance affected in a pipelined processor?

Calculate Atomic Packing Fraction (APF) for FCC and HCP structures, and show that these are the most closely packed structures.

What are intrinsic and extrinsic semiconductors? Show that in the intrinsic semiconductors, Fermi level lies exactly in the middle of bottom of conduction band and top of valence band.

A pair of transistors operates in a class B push-pull circuit with a power supply of voltage V_{\text{cc}} = 12\text{ V}. If the turns ratio of the output transformer is 2 and the load resistance R_{\text{L}} is 6\ \Omega, calculate the maximum output power and maximum dc power supplied. Hence find efficiency of the circuit. Neglect transformer winding resistance.

What are type I and type II superconductors? Give examples. Discuss and compare Meissner effect and perfect diamagnetic behaviour for type I and type II superconductors.