UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Calculate the inertia tensor for a rigid body consisting of three particles of masses 3\text{ gm}, 1\text{ gm}, 2\text{ gm} located at (1, -1, 2)\text{ cm}, (1, 0, 2)\text{ cm}, (-2, 1, 0)\text{ cm} respectively.

A particle of mass m moves under the action of a central force whose potential is V(r) = Kmr^4 (K > 0). Calculate the kinetic energy for which the orbit will be circle of radius R, about the origin.

A hoop is rolling down on an inclined plane without slipping. Find its velocity at the bottom of the inclined plane.

Two spaceships are approaching each other as depicted below.

Spaceship 1 is moving with a velocity \vec{\text{u}}_1 = 0\cdot 7\text{c }\hat{\text{x}}, while spaceship 2 is moving with a velocity \vec{\text{u}}_2 = -\,0\cdot 8\text{c }\hat{\text{x}} as measured by the stationary observer on Earth. Here, c is the velocity of light in vacuum and \hat{\text{x}} is the unit vector along the x-axis.

(i) What, according to Galilean addition of velocity, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(ii) What, according to relativistic addition of velocities, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(iii) Spaceship 1 emits a light signal towards spaceship 2. The observer on spaceship 2 measures the frequency of the light signal to be 10^{16}\text{ Hz}. What was the original frequency of the light signal ?

Consider the motion of a particle of mass m in central force field \text{f(r)} = -\,\text{k}/\text{r}^2, where k is a constant and r is the distance from the force centre to the particle. Show that for the given system the vector \vec{\text{A}} = \vec{\text{p}} \times \vec{\text{L}} - \text{mk} \, \frac{\vec{\text{r}}}{\text{r}} is conserved. Using this vector and the fact that \vec{\text{A}} \cdot \vec{\text{L}} = 0, derive the orbit equation for the particle.

A small block of mass m slides without friction down a wedge-shaped block of mass M and opening angle \alpha as depicted below.

The wedge-shaped block itself slides along the horizontal frictionless floor along the +\text{ve x-direction}. Find the horizontal acceleration \ddot{\text{X}} = \text{d}^2\text{X}/\text{dt}^2 of the wedge-shaped block by the Lagrangian method and using the displacements \text{X} and \text{S} of the blocks as generalized coordinates.

Consider a particle of mass m moving in a potential field of the form \text{V}(\vec{\text{r}}) = \text{V}_0(\text{x}^2 + \text{y}^2), where \text{V}_0 is a constant. What are the conserved physical quantities for the particle ?

Consider the force-free motion of a symmetrical rigid body with z-axis as its axis of symmetry. Write down Euler's equations. Integrate them and obtain the solutions. On the basis of the obtained solutions show that the total angular velocity \vec{\omega} of such a rigid body is constant in magnitude and precesses about the z-axis with a constant frequency, say \Omega. Determine \Omega.

A particle of rest mass \text{m}_0 is moving in a stationary frame K with a velocity \vec{\text{u}} making an angle \theta with the x-axis as shown in the figure.

Find the corresponding angle \theta' made by the velocity vector of the particle with the \text{x}' axis in the frame \text{K}' that is moving with respect to K along the x-direction at a constant speed V.

A comet of mass m moves towards the Sun with initial velocity v_0. The mass of the sun is M and its radius is R. Find the total cross-section \sigma for striking the Sun. Take the Sun to be at rest and ignore all other bodies.

A block of mass m is sliding on a wedge of mass M as shown in the figure below. The wedge can slide on the horizontal table. Find the equation of motion.

On a space-time diagram after the Lorentz transformation with given v we get new tilted axis ct' and x'. For a point on the ct' axis which is at the (geometrical) distance \delta on the diagram measured from the origin, find the values of t' and t.

A rigid body is rotating under the influence of an external torque (N) acting on it. If T is the kinetic energy and \omega is the angular velocity, show that \dfrac{dT}{dt} = N \cdot \omega in the principal axes system.

A pion at rest decays into a muon and a neutrino.

(i) Find the energy of the outgoing muon in terms of the two masses m_\pi and m_\mu (assuming m_\nu = 0). (ii) Find the velocity of muon.

During take-off, an aircraft accelerates horizontally in a straight line at a rate A. A small bob of mass m is suspended on a string attached to the roof of the cabin, and a hydrogen balloon (total mass m) is tethered to the floor by a string. For each, determine the tension in the string and the equilibrium angle \theta between string and vertical. Draw a neat diagram to explain your answer.

Obtain the Lorentz transformations for the components of momentum-energy four vector.

(i) Define the angular velocity \omega of a rigid body rotating about some axis.

(ii) Then derive the relation \vec{v} = \vec{\omega} \times \vec{r} for the velocity of a point in the body with position vector \vec{r} relative to an origin on the axis. Draw a diagram explaining all the vectors.

(i) Explain Coriolis force. Compare the qualitative effects of the Coriolis forces in the two geographical hemispheres of the Earth for rivers flowing east. (ii) With schematic, explain Foucault's pendulum.

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.

Two spaceships approach each other, each moving with the same speed as measured by a stationary observer on the Earth. Their relative speed is 0{\cdot}7c. Determine the velocity of each spaceship as measured by the stationary observer on the Earth.