CSS 2022 Physics Paper I

A particle of unit mass moves in a potential V(x) = ax^2 + b/x^2, where a and b are positive constants. Find the angular frequency of small oscillations about the equilibrium position.

A hollow spherical shell carries a charge density ρ = k/r^2 in the region a ≤ r ≤ b. Find the electric field in the regions (i) r < a, (ii) a < r < b, and (iii) r > b.

A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. Determine its angle of projection.

Assume that a star has uniform density. Show that the gravitational pressure P is proportional to V^(-4/3) where V is the volume of the star.

Determine the equation of motion for the masses attached to the string of an Atwood machine using Lagrangian methods.

Q cm³/s of water flows through a horizontal tube of uniform bore of radius r and length L. Another tube of half the length but radius 2r is connected in parallel to the same pressure head. What will be the total quantity of water flowing per second through the combination?