Derive the relativistic expression for kinetic energy by considering mass variation with velocity. Hence, establish the relation between momentum (p) and energy (E) for a relativistic particle; \frac{dE}{dp}=v.

What do you understand by length contraction? Calculate the percentage length contraction of a rod moving with a velocity 0.8c in a direction at 60^\circ with respect to its own length.

Determine the location of the centre of mass of a uniform solid hemisphere of radius R and mass M from the centre of its base.

A rocket of mass m_1 + m_2 is launched with a velocity whose horizontal and vertical components are u_x and u_y. At the highest point in its path, the rocket explodes into two parts of masses m_1 and m_2 that separate in a horizontal direction in the original plane of motion. Show that the fragments strike the ground at a distance apart given by D = \frac{u_y}{g} \sqrt{\frac{2(m_1 + m_2)K}{m_1 m_2}} where K is the kinetic energy produced by the explosion. (Neglect air resistance, the mass of explosive, any spinning motion of the fragments and assume that g is constant)

A shaft of diameter 8\,\mathrm{cm} and length 5\,\mathrm{m} is transmitting power of 8\,\mathrm{kW} at 300 revolutions per minute. If the coefficient of rigidity of the material of the shaft be 8 \times 10^{11}\,\mathrm{dynes/cm^2}, then calculate the relative shift between the ends of the shaft.

A rubber cord 1\ \mathrm{mm} in diameter and 1\ \mathrm{m} long is fixed at one end and a weight of 1\ \mathrm{kg} is attached to the other end. If the Young's modulus of rubber is 0.05 \times 10^{11}\ \mathrm{dynes\,cm^{-2}}, then find the period of the vertical oscillations of the weight.

Obtain expressions for the moment of inertia of a solid cone about its \begin{qroman}

(i) Vertical axis and

(ii) axis passing through the vertex and parallel to its base.

Suppose the position of a particle of mass m on the xy-plane is \bar{r} = (x, y, z) = (r \cos\phi, r \sin\phi, 0) Here r and \phi are functions of time t. Find l_z, the z-component of \vec{l}, where \vec{l} is the angular momentum of the particle. Further, show that the areal velocity is equal to the angular momentum divided by 2m. Is areal velocity constant?

A rocket starts vertically upwards with speed v_0. Then define its speed v at a height h in terms of v_0, h, R (radius of Earth) and g (acceleration due to gravity on Earth's surface). Also calculate the maximum height attained by a rocket fired with a speed of 90% of the escape velocity.

The moment of inertia of the CO molecule is 1\cdot 46 \times 10^{-46}\text{ kg-m}^2. Calculate the energy (in eV), and the angular velocity in the lowest rotational energy level of the CO molecule.

Explain the rotation of a free rigid body and show that a rigid body rotating in any manner about a fixed point has two constants of motion, L^2 and T. Here \vec{L} is angular momentum and T is kinetic energy.

A superconducting material has a critical temperature of 3\cdot 7~\text{K} and a magnetic field of 0\cdot 0306~\text{tesla} at 0~\text{K}. Find the critical field at 2~\text{K}.

Explain the Meissner effect in superconductivity. Obtain an expression for London penetration depth of magnetic field for a superconductor.

A silicon semiconductor sample at T=300\,\mathrm{K} having cross-sectional area of 0.5\,\mu\mathrm{m}^2 has a pentavalent donor doping profile given by C(x)=5\times10^{16}e^{-x/L_n}\,\mathrm{cm^{-3}}. Given, the mobility of the electrons in the sample is 1250\,\mathrm{cm^2\,V^{-1}\,s^{-1}} and the diffusion length of the electrons, L_n, is 4\,\mu\mathrm{m}. Calculate the diffusion current in the sample at distance x=2\,\mu\mathrm{m}.

Obtain the Boolean expression for the output Y in the given logic circuit :

Simplify the expression and show that the circuit is equivalent to an AND gate with inputs A and B.

Deduce the Miller indices of the close-packed planes of atoms in the f.c.c. lattice.

An n-channel silicon JFET has donor concentration of 2 \times 10^{21}/\text{m}^3 and a channel width of 4~\mu\text{m}. If the dielectric constant of silicon is 12, find the pinch-off voltage. If the FET operates with a gate-source voltage of -2~\text{V}, what is the saturation voltage V_{\text{Dsat}}?

Explain the use of an op-amp as a differentiator and an integrator using proper circuit diagram. Write the condition to work as good differentiator and integrator.

A transistor having \alpha = 0\cdot 975 and a reverse saturation current I_{\text{CO}} = 10~\mu\text{A} is operated in CE configuration. What is \beta for this configuration? If the base current is 250~\mu\text{A}, calculate the emitter current and the collector current.

Sketch the cross-sectional structure of an enhancement mode MOSFET and explain its principles of operation with the help of its output characteristics.