UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Discuss that the motion of a particle in a central field gets restricted to a plane provided its angular momentum is nonzero. What will happen if the angular momentum is zero?

A particle of mass m is moving with the velocity \vec{v} on the surface of the earth. Discuss the nature of its motion in both the hemispheres due to the Coriolis force \vec{F} = -2\,m\,\vec{\omega} \times \vec{v}.

Solve Euler's equation of motion for a rigid body to show that the torque-free motion of a spherical top is a uniform rotation about an axis fixed in space.

The Lagrangian for a particle has the form L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - U(r) Obtain its form in plane polar coordinates and then get expression for the Hamiltonian using appropriate transformation.

Find the velocity and momentum of a particle having rest mass m_0 and kinetic energy equal to two times of its rest mass energy.

Describe briefly the Michelson-Morley experiment and discuss the significance of its null result in the context of the special theory of relativity.

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?

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?

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.

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.

Deduce the minimum energy of a gamma ray photon (in MeV), which can cause electron-positron pair production.

Discuss the torque-free motion of a symmetric rigid body and hence estimate the period of precession of Earth's polar axis. [Use \frac{I_3 - I_1}{I_3} = 0.33, where I_1, I_2 and I_3 are the moments of inertia about the three principal axes of the Earth]

State Hamilton's principle for a system of particles. If I and L represent the action integral and the Lagrangian function, respectively, write down the mathematical form of Hamilton's principle and explain clearly the significance of the same.

A particle of rest mass M moving at a velocity u collides with a stationary particle of rest mass m. If the particles stick together, show that the speed of the composite ball is equal to u\alpha M/(\alpha M + m), where \alpha = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} .

Derive the law of addition of relativistic velocities. Use it to prove that under the Lorentz transformation no two velocities can add upto more than the value of the speed of light.

Using the concept of D'Alembert's principle, show that the generalized force can be defined as Q_j = \sum_i \bar{F}_i \cdot \frac{\partial \bar{r}_i}{\partial q_j} , where \bar{r}_i is the Cartesian coordinate of the i^{\text{th}} particle experiencing external force \bar{F}_i and q_j stands for the generalized coordinate. Discuss the significance of the above expression.

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.

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.