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. Youngs 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.

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 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 capillary tube having 1.0\,\mathrm{mm} diameter, 20\,\mathrm{cm} in length is fitted horizontally to a vessel in which alcohol is kept fully up to the neck. Density of alcohol is 8 \times 10^{2}\,\mathrm{kg/m^3}. The depth of the centre of the capillary tube below the surface of alcohol is 40\,\mathrm{cm}. Find the amount of alcohol that will flow out of the capillary tube in 10 minutes. Coefficient of viscosity of alcohol is 0.0012\,\mathrm{N\,s/m^2}.

(i) Calculate the mass and momentum of a proton of rest mass 1.67\times10^{-27}\,\mathrm{kg} moving with a velocity of 0.8c, where c is the velocity of light. If it collides and sticks to a stationary nucleus of mass 5\cdot0\times10^{-26}\,\mathrm{kg}, find the velocity of the resultant particle.

(ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling.

Obtain expressions for the moment of inertia of a solid cone about its \begin{qroman}

(i) Vertical axis and

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

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.

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.

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.

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.

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.

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?

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).

(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.

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.

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.