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.
Write down the Euler equations for torque-free motion of a rigid body. Solve these equations to find precessing motion for a symmetric top.
Calculate the effective gravitational acceleration at the equator due to the Earth's rotation. [\text{R}_{\text{earth}} = 6400\text{ km}]
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.
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.
Consider two inertial frames \text{S} and \text{S}'. \text{S}' is moving along \text{y}-direction with a constant speed \text{v}_0. Find how the first order partial derivatives with respect to \text{x}, \text{y}, \text{z} and \text{ct} are connected in these two inertial frames. Show that the D'Alembertian, \nabla^2 - \frac{1}{\text{c}^2}\frac{\partial^2}{\partial \text{t}^2} is invariant under the Lorentz transformation.
Two particles each of rest mass 1\text{ gm} collide head-on with the speed 0\cdot 8\text{c} to stick together at rest. Find the mass of the final composite.
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.
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.
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.
Compare the working of FET and MOSFET with their structure, I-V curve and transfer characteristics.
(i) What is the origin of paramagnetism ? Discuss the temperature dependence of paramagnetic susceptibility. (ii) Calculate the diamagnetic susceptibility of atomic Hydrogen in the ground state at STP. Assume the mean square distance of electronic charge distribution of atomic Hydrogen from the nucleus r^2 = 3 a_0^2, a_0 being the radius of the first Bohr orbit of Hydrogen.
(i) Prove that \overline{AB + \bar{A} + AB} = 0.
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.
Consider the planes with indices (1\ 0\ 0) and (0\ 0\ 1); the lattice is f.c.c., and the indices refer to the conventional cubic cell. What are the indices of these planes when referred to the primitive axes : \vec{a}_1 = \frac{a}{2}(\hat{x}+\hat{y}), \vec{a}_2 = \frac{a}{2}(\hat{y}+\hat{z}) and \vec{a}_3 = \frac{a}{2}(\hat{z}+\hat{x})?
(ii) Draw a simplified logic circuit for the following Boolean expression : Y = (A + B)\overline{AB}
State the Barkhausen criterion for sustained oscillations of a feedback circuit. Explain the operation of a Hartley oscillator using proper circuit diagram.
State the characteristics of an ideal Op-Amp. Explain the use of an Op-Amp as a summing amplifier and a differentiator.
