Using the concept of the operator equation for a given vector quantity of a rotating body, show that \vec{v}_s = \vec{v}_r + \vec{\omega} \times \vec{r} where \vec{v}_s and \vec{v}_r are the velocities of the particle relative to space and rotating sets of axes while \vec{\omega} is the angular velocity of the earth relative to the inertial system. Obtain the expression for the Coriolis force for such a moving system of mass m.
State Hamilton's principle for the motion of a monogenic system. Use the calculus of variation to deduce Lagrange's equation that follows from Hamilton's principle. Explain carefully all the terms used in the derivation.
Using D'Alembert's principle, show that the following relation can be obtained for a system of particles under generalized coordinates \sum_i \vec{F}_i \cdot \delta \vec{r}_i = \sum_j Q_j \delta q_j with Q_j = \sum_i \vec{F}_i \cdot \frac{\partial \vec{r}_i}{\partial q_j}. What is the significance of Q_j ?
Starting with Newton's second law of motion, establish D'Alembert's principle and discuss its significance.
Express Lagrange's equation of motion for the cyclic coordinate q_j and show that the result leads to the general conservation theorem for the generalized momentum coordinates.
A man has a mass of 100\text{ kg} on earth. When he is on the space craft an observer from the earth registers his mass as 102\text{ kg}. Determine the speed of the space craft.
Using OP-AMPs design an analog circuit to solve the following differential equation : \frac{d^2 v}{dt^2} + k_1 \frac{dv}{dt} + k_2 v = v_1(t) with \frac{dv_{(0)}}{dt} = 1\cdot 5\text{ V}; v_{(0)} = 2\cdot 0\text{ V}.
(i) What are intrinsic and extrinsic semiconductors ? Explain. (ii) How are n-type and p-type semiconductors obtained ?
Using band theory of solids, explain whether the effective mass of an electron can be positive, negative as well as infinity. Explain the significance of negative mass.
How are the band structures of insulators and semiconductors similar ? How are they different ?
Explain the basic structure and operation of an n-channel FET. How FET acts as an amplifier ? Explain.
What type of interaction takes place in the following reactions and justify your answer. (i) \mu^- \longrightarrow e^- + u_e + \bar{\nu}_\mu (ii) p + \mu^- \longrightarrow n + u_\mu
What are the reactor materials used in design and construction of a nuclear reactor and their important functions ?
What are magic numbers in an atomic nucleus and why are they so called ? How is it explained using nuclear shell model ?
Explain the source of energy in stars using proton-proton thermonuclear reaction cycles.
Find the density of {}^{12}_6 C nucleus. (Given, nuclear radius of carbon = 2\cdot 7 \times 10^{-15} \text{ m})
What are quarks ? Give any four properties. Give quark composition of the following particles : p,\quad n,\quad \Lambda^\circ,\quad \Pi^+,\quad K^+
What is the mechanism of emission of light in fluorescent lamps and in painted signboards ? Explain.
Give the elementary theory of NMR. Explain the two different relaxation processes.
Discuss the vibrational spectra of a diatomic molecule treating it as an anharmonic oscillator.
