UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?

Calculate the percentage contraction in the length of a rod in a frame of reference, moving with velocity 0.8c in a direction

(i) parallel to its length and

(ii) at an angle of 30^\circ with its length. What is the orientation of the rod in the moving frame of reference in case (ii)?

How does one obtain the angular velocity of the Earth about the North Pole with respect to a fixed star as 7.292 \times 10^{-5}\,\mathrm{sec}^{-1}? Explain your method of calculating the above value.

Show that the moment of inertia of a circular disc of mass M and radius R about an axis passing through its centre and perpendicular to its plane is \frac{1}{2}MR^2.

Using Poiseuille's formula, show that the volume of a liquid of viscosity coefficient \eta passing per second through a series of two capillary tubes of lengths l_1 and l_2 having radii r_1 and r_2 is obtained as Q=\frac{\pi p}{8\eta\left[\frac{l_1}{r_1^4}+\frac{l_2}{r_2^4}\right]}. where p is the effective pressure difference across the series.

Show that the differential scattering cross-section can be expressed as \sigma(\theta)=\frac{s}{\sin\theta}\left|\frac{ds}{d\theta}\right|, where s is the impact parameter and \theta is the scattering angle.

Write down Poiseuille's formula and mention its limitations in analysing the flow of a liquid through a capillary tube.

Prove mathematically that the addition of any velocity of a particle to the velocity of light in free space merely reproduces the velocity of light in free space only.

Show that the rest mass energy of an electron is 0.51\,\mathrm{MeV}. (use the standard values of the physical parameters).

Draw a neat diagram to explain the scattering of an incident beam of particles by a centre of force.

Define coefficients of viscosity and kinematic viscosity of a fluid. What are Poise and Stokes?

Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms.

The density inside a solid sphere of radius a is given by \rho = \frac{\rho_0 a}{r}, where \rho_0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.

If I' and I be the Moments of Inertia of a body about an axis passing through an arbitrary origin and about a parallel axis through the centre of mass respectively, show that I' = MR^2 + I, where \vec{R} is the position vector of the centre of mass with respect to the arbitrary origin and M is the mass of the body.

Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression?

A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time period of the motion.

Consider a rigid body rotating about an axis passing through a fixed point in the body with an angular velocity \vec{\omega}. Determine the kinetic energy of such a rotating body in a coordinate system of principal axis. If the earth suddenly stops rotating, what will happen to the rotational kinetic energy? Comment in detail.

A body turns a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.

A sphere of radius R moves with velocity \vec{u} in an incompressible, non-viscous ideal fluid. Calculate the pressure distribution over the surface of the sphere. Do you think that a force is necessary to keep the sphere in uniform motion?

A mirror is moving through vacuum with a relativistic speed v in the x-direction. A beam of light with frequency \omega_i is normally incident (from x=\infty) on the mirror.

(i) What is the frequency of the reflected light expressed in terms of \omega_i, c and v?

(ii) What is the energy of each reflected photon?