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.
Consider a light beam passing through a horizontal column of water moving with a velocity 'v'. Determine the speed u of the light measured in the lab frame when the beam travels in the same direction as the flow of the water. (Speed of water = v (lab frame), and refractive index = n)
A particle of mass m rests on a smooth plane. The plane is then raised to an inclination angle \theta at a constant rate \alpha (\theta = 0 at t = 0), causing the particle to move down the plane, as shown in the figure below.
Write down the Lagrangian and determine the equations of motion. Solve the resulting equation for "r" to obtain the expression for r(t).
(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}}.
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?
A three-particle system consists of masses \text{m}_i,\ i = 1, 2, 3 and their respective coordinates (\text{x}_i, \text{y}_i, \text{z}_i) : m_1 = 3m,\ (x_1, y_1, z_1) = (a, 0, a), m_2 = 4m,\ (x_2, y_2, z_2) = (a, a, -a), m_3 = 2m,\ (x_3, y_3, z_3) = (-a, a, 0), where m and a are positive constants. Calculate the inertia tensor of the system.
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).
How fast and in what direction must galaxy A be moving if an absorption line found at wavelength 550\text{ nm} (green) for a stationary galaxy is shifted to 450\text{ nm} for galaxy A and how fast and in what direction is galaxy B moving if the same line is shifted to 700\text{ nm} for it ?
A particle moves in an elliptical orbit in an inverse-square-law central force field. If the ratio of the maximum angular velocity to the minimum angular velocity of the particle in the orbit is n, then show that the eccentricity of the orbit is \varepsilon = \frac{\sqrt{n} - 1}{\sqrt{n} + 1}.
A mouse of mass m jumps on the outside edge of a freely turning ceiling fan of rotational inertia I and radius R. By what ratio does the angular velocity change ?
(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.
A rod of length l_0 is kept at rest in x'\,y' plane of its rest frame making an angle \theta_0 with x' axis. What is the length and orientation of the rod in a laboratory frame (x,y) in which the rod moves to the right with velocity v?
Show that the wave equation for the propagation of electromagnetic scalar potential \phi(x, y, z, t) \left[\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right] \phi(x, y, z, t) = 0 remains invariant under Lorentz transformation.
A particle of mass m is constrained to move on a plane curve xy=c with c>0, under gravity, where y-axis is vertical. Construct the Lagrangian of the system and obtain the Euler-Lagrange equation of motion.
A particle moving in a central force field located at r=0, describes a spiral, r=e^{-\theta}. Find the force law.
Consider a homogeneous cube of total mass M and side a. Taking the origin at one corner of the cube and axes along the edges of the cube, construct the moment of inertia tensor. Calculate principal moment of inertia.
Calculate the first relativistic correction to the kinetic energy of a particle with rest mass m_0 and speed v.
In special theory of relativity, the Hamiltonian of a free particle with rest mass m_0 is given by \sqrt{p^2 c^2 + m_0^2 c^4}. Obtain the Lagrangian of the system using Legendre transformation.
A water drop of radius 0.04\ \mathrm{mm} is falling through air. If the coefficient of viscosity for air is 1.8 \times 10^{-4} poise, find its terminal velocity. If 100 such drops coalesce, what will be the new terminal velocity?
