PYQ Bank

What are Einstein's A and B coefficients ? Establish a relation between them.

Explain the Fraunhofer diffraction at a single slit and obtain the condition for minima. Derive the expression for resolving power of a grating.

Distinguish between Fresnel and Fraunhofer classes of diffraction. Show that the area of each Fresnel half-period zone is same.

Two identical relativistic particles of rest mass m and kinetic energy T collide head-on. What is the relative kinetic energy, i.e., the kinetic energy T' of one in the rest frame of the other?

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.

A particle is moving in a central force field on an orbit given by r = ke^{\alpha\theta}, where k and \alpha are positive constants, r is the radial distance and \theta is the polar angle.

(i) Find the force law for the central force field.

(ii) Find \theta(t).

(iii) Find the total energy.

Starting with Newton's second law of motion, establish D'Alembert's principle and discuss its significance.

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 ?

Calculate the moment of inertia of a solid cone of mass M, height h, vertical half-angle \alpha and radius of its base R, about an axis passing through its vertex and parallel to its base.

A rigid body is spinning with an angular velocity of 4\ \mathrm{rad\,s^{-1}} about an axis parallel to the direction (4\hat{\jmath}-3\hat{k}) passing through the point A with \overrightarrow{OA}=2\hat{\imath}+3\hat{\jmath}-\hat{k}, where O is the origin of the coordinate system. Find the magnitude and direction of the linear velocity of the body at point P with \overrightarrow{OP}=4\hat{\imath}-2\hat{\jmath}+\hat{k}.

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.

Define a conservative field. Determine if the field given below is conservative in nature: \vec{E}=c\left[y^2\hat{i}+(2xy+z^2)\hat{j}+2yz\hat{k}\right]\,\mathrm{V/m} where c is a constant.

Consider a uniform half-sphere of radius R and mass M. The half-sphere is supported by a frictionless horizontal plane as shown in the figure. The half-sphere lies in the region z<0.

Find the centre of mass of the half-sphere.

A projectile of mass M explodes, while in flight, into three fragments. One fragment of mass m_1=M/2 travels in the original direction of the projectile. Another fragment of mass m_2=M/6 travels in the opposite direction and the third fragment of mass m_3=M/3 comes to rest. The energy E, released in the explosion, is 5 times the kinetic energy of the projectile at explosion. What are the velocities of the fragments?

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.

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 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.

Simplify the logical expression AB + \bar{A}\bar{B} + ABC using a Karnaugh map.

Explain the origin of energy band formation in solids. Show that in nearly free electron approximation, the energy band gap is 2|V_G|, where V_G is Fourier transform of periodic potential seen by the valence electrons.

Find an expression for lattice specific heat of solids, and its low and high temperature limits. What is Debye temperature?