UPSC Physics Optional PYQs (CSE & IFoS)PYQs

A homogeneous right triangular pyramid with the base side a and height \frac{3a}{2} is shown below. Obtain the moment of inertia tensor of the pyramid:

A light rod of length 100\,\mathrm{cm} is suspended from the ceiling, horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of steel and its cross-section is 0.05\,\mathrm{sq.\ cm} and the other is of brass of cross-section 0.1\,\mathrm{sq.\ cm}. Find the position along the rod at which a weight may be hung to produce

(i) Equal stresses in both the wires,

(ii) Equal strain in both the wires. Young's modulus of elasticity of brass and steel are 1.0 \times 10^{11}\,\mathrm{N/m^2} and 2.0 \times 10^{11}\,\mathrm{N/m^2} respectively.

A body of mass m at rest splits into two masses m_1 and m_2 by an explosion. After the split the bodies move with a total kinetic energy T in opposite direction. Show that their relative speed is \sqrt{\frac{2Tm}{m_1m_2}}.

An electron is moving under the influence of a point nucleus of atomic number Z. Show that the orbit of the electron is an ellipse.

An observer detects two explosions, one that occurs near him at a certain time and another that occurs 2\,\mathrm{ms} later 100\,\mathrm{km} away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer?

A rocket starts vertically upwards with speed v_0. Then define its speed v at a height h in terms of v_0, h, R (radius of Earth) and g (acceleration due to gravity on Earth's surface). Also calculate the maximum height attained by a rocket fired with a speed of 90% of the escape velocity.

What do you understand by length contraction? Calculate the percentage length contraction of a rod moving with a velocity 0.8c in a direction at 60^\circ with respect to its own length.

Derive the relativistic expression for kinetic energy by considering mass variation with velocity. Hence, establish the relation between momentum (p) and energy (E) for a relativistic particle; \frac{dE}{dp}=v.

A shaft of diameter 8\,\mathrm{cm} and length 5\,\mathrm{m} is transmitting power of 8\,\mathrm{kW} at 300 revolutions per minute. If the coefficient of rigidity of the material of the shaft be 8 \times 10^{11}\,\mathrm{dynes/cm^2}, then calculate the relative shift between the ends of the shaft.

Determine the location of the centre of mass of a uniform solid hemisphere of radius R and mass M from the centre of its base.

Obtain expressions for the moment of inertia of a solid cone about its

(i) Vertical axis and

(ii) axis passing through the vertex and parallel to its base.

A rubber cord 1\ \mathrm{mm} in diameter and 1\ \mathrm{m} long is fixed at one end and a weight of 1\ \mathrm{kg} is attached to the other end. If the Young's modulus of rubber is 0.05 \times 10^{11}\ \mathrm{dynes\,cm^{-2}}, then find the period of the vertical oscillations of the weight.

(i) How does Reynolds number help in the study of fluid motion?

(ii) In a horizontal pipeline of uniform area of cross-section, the pressure falls by 5\ \mathrm{N\,m^{-2}} between two points separated by a distance of 1\ \mathrm{km}. Calculate the change in kinetic energy per kg of oil flowing at these points. Density of oil = 800\ \mathrm{kg\,m^{-3}}.

(i) A reference frame S' moves with respect to rest frame S with a uniform velocity v parallel to x-direction. Show from Lorentz transformation that two events simultaneous (t_1=t_2) at different positions (x_1\ne x_2) in S frame are not in general simultaneous in S' frame.

(ii) The mean life of \pi meson is 2\times10^{-8}\,\mathrm{s}. Calculate the mean life of a meson moving with a velocity of 0.8c, where c is the velocity of light.

Two \beta-particles A and B emitted by a radioactive source R travel in opposite directions, each with a velocity of 0.9c with respect to the source. Find the velocity of B with respect to A (Here c is the velocity of light).

Where do you find the applications of gyroscope? A top of mass 0.200\ \mathrm{kg} is made up of a thin disc of radius 0.12\ \mathrm{m}. It is pierced in the centre and a pin of negligible mass is mounted normal to its plane. The pivot under the disc is 0.03\ \mathrm{m} long. The top is made to spin with its axis making an angle \theta = 20^\circ with the vertical and a precessional angular speed of 2\ \mathrm{rad/s}. Calculate the angular speed with which it spins.

(i) What is central force? Give two examples of the central force.

(ii) Show that the angular momentum \vec{L} of the particle in a central force field is a constant of motion.

Show that the cross-section for elastic scattering of a point particle from an infinitely massive sphere of radius R is \dfrac{R^2}{4}. What is the inference of this result?

Write down Euler's dynamical equations of motion (no derivation) of a rigid body about a fixed point under the action of a torque. Show that the kinetic energy of the torque-free motion is constant.

(i) Find the moments of inertia of rigid diatomic molecule about different axes of symmetry through the centre of mass.

(ii) A proton is 1837 times heavier than an electron. Find the centre of mass of hydrogen atom.