UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Consider three inertial frames of reference O, O' and O''. Let O' move with a velocity V with respect to O and O'' move with a velocity V' with respect to O'. Both velocities are in the same direction. Write down the transformation equations relating x, y, z, t with x', y', z', t' and also those relating x', y', z', t' with x'', y'', z'', t''. Hence obtain the relations between x, y, z, t and x'', y'', z'', t''. (The direction of velocity is chosen along the x-axis as per convention)

Consider two inertial frames S and S'. S is at rest and S' is moving along x - x' direction with constant speed v. (i) Find how different components of momentum four-vector in S and S' frames are related. (ii) Show that p^\mu p_\mu is Lorentz invariant.

The moment of inertia tensor of a cube of mass M and side a is given by the matrix I = \frac{M a^2}{12} \begin{bmatrix} 8 & -3 & -3 \\ -3 & 8 & -3 \\ -3 & -3 & 8 \end{bmatrix}. Calculate the principal moments of inertia of the cube.

State and explain the Hooke's law of elasticity. Briefly discuss the features of stress-strain diagram for the behaviour of a wire undergoing increasing stress.

A charged \pi-meson with rest mass of 273 m_e at rest decays into a neutrino and a \mu-meson of rest mass 207 m_e. Find the kinetic energy of the \mu-meson and the energy of the neutrino. (m_e is the rest mass of the electron)

A galaxy in the constellation Ursa Major is receding from the Earth at 15000\text{ km/s}. If one of the characteristic wavelengths of light emitted by the galaxy is 550\text{ nm}, what is the corresponding wavelength measured by astronomers on the Earth?

Briefly discuss the Kepler's laws of planetary motion.

Show that the escape velocity V_e on the surface of the Earth is given by V_e = \sqrt{2gR}, where g = 9.8\text{ m/s}^2 and R is the radius of the Earth.

Two satellites A and B of same mass are orbiting the Earth at altitudes R and 5R, respectively, where R is the radius of the Earth. Assuming their orbits to be circular, calculate the ratios of their kinetic and potential energies.

Define Coriolis force. A particle is dropped from a height h vertically at the northern hemisphere at latitude \theta. Find its deflection from the vertical line due to Coriolis force.

Show that the kinetic energy of a system of n particles is given by T = \frac{1}{2} M V_{\text{cm}}^2 + \frac{1}{2} \sum_{i=1}^n m_i {V'_i}^2 where M is the total mass, V_{\text{cm}} is the velocity of the centre of mass, V'_i is the velocity of the particles about the centre of mass and m_i is the mass of the i\text{th} particle.

The Lagrangian of a system is given by L (\vec{r}, \vec{v}) = \frac{1}{2} m (\vec{v} + \vec{a})^2 + \vec{b} \cdot \vec{v} - \vec{c} \cdot \vec{r} where, \vec{a}, \vec{b} and \vec{c} are constant vectors. Construct the Hamiltonian of the system and find the canonical equations of motion.

Construct the Lagrangian of a simple pendulum whose string is replaced by a spring with spring constant k and rest length l_0 oscillating in a vertical x - y plane. Find the equations of motion.

A particle of mass m\text{ kg} having an initial velocity V_0 is subjected to a retarding force proportional to its instantaneous velocity. Obtain the expression for the velocity and position of the particle as a function of time.

Show that the angular momentum of a rigid body consisting of n particles of masses m_i, i = 1, 2, 3, \dots, n, rotating with an instantaneous angular velocity \mathbf{\omega} about an axis passing through the origin O of the coordinate system OXYZ is given by \mathbf{L} = \mathbf{I} \cdot \mathbf{\omega}, where \mathbf{I} is known as the inertia tensor.

Define generalized coordinates. How many generalized coordinates are required to describe the dynamics of (i) a free rigid body in 3 dimension ? (ii) a solid cylinder rolling in an inclined plane without slipping ?

Find the equation of orbit for a particle of mass m, moving in the influence of central force F(r) in terms of u \left(\equiv \frac{1}{r}\right).

(i) Determine the input and output impedances of the amplifier in given figure. The op-amp datasheet gives Z_{\text{in}} = 2\text{ M}\Omega, Z_{\text{out}} = 75\ \Omega and A_{\text{ol}} = 200,000 (open loop voltage gain). (ii) Find the closed-loop voltage gain.

With a neat circuit diagram, explain the working of Wien-Bridge oscillator.

Consider the BJT amplifier circuit given below with the following parameters : d.c. current gain \beta_{\text{dc}} = a.c. current gain \beta_{\text{ac}} = 100 \text{I}_\text{B} = 20\ \mu\text{A} for \text{V}_\text{B} = 0\cdot 7\text{ V}, and \text{i}_\text{b} = \pm 2\ \mu\text{A} for \text{v}_\text{i} = \pm 10\text{ mV}. Determine the values of \text{V}_{\text{CE}}, input impedance and voltage gain \text{A}_\text{v}.