Show that the kinetic energy of a system of n particles is given by T = \frac{1}{2} M V_{\text{cm}}^2 + \frac{1}{2} \sum_{i=1}^n m_i {V'_i}^2 where M is the total mass, V_{\text{cm}} is the velocity of the centre of mass, V'_i is the velocity of the particles about the centre of mass and m_i is the mass of the i\text{th} particle.
Briefly discuss the Kepler's laws of planetary motion.
Two satellites A and B of same mass are orbiting the Earth at altitudes R and 5R, respectively, where R is the radius of the Earth. Assuming their orbits to be circular, calculate the ratios of their kinetic and potential energies.
(i) Prove that the separation of two colliding particles is same, when observed in centre of mass and laboratory systems. (ii) Determine the kinetic energy of a thin disc of mass 0\cdot5\text{ kg} and radius 0\cdot2\text{ m} rotating with 100 rotations per second around the axis passing through its centre and perpendicular to its plane.
Calculate the gravitational self-energy of the Earth. Given : Mass of Earth M_e = 6 \times 10^{24}\text{ kg} and the Radius of Earth R_e = 6\cdot4 \times 10^6\text{ m}
(i) Derive the expressions for gravitational potentials at a point (I) outside the spherical shell, (II) inside the spherical shell. (ii) Calculate the escape velocity of a body of mass 10\text{ kg} from the surface of Moon \left( g_{\text{Moon}} = \frac{1}{6} g_{\text{Earth}} \right). Mass of Moon = 7\cdot3 \times 10^{22}\text{ kg} Radius of Moon = 1\cdot7 \times 10^6\text{ m}
(i) The quantities of rotatory motion are analogous to those of translatory motion. Write the corresponding equations of translatory and rotatory motion. (ii) Describe the theorems of perpendicular and parallel axes in case of a plane lamina.
A force \vec{F} is given by \vec{F} = x^2 y \hat{x} + z y^2 \hat{y} + x z^2 \hat{z}. Determine whether or not the force is conservative.
Define streamline flow of a fluid. Using the equation of continuity for an isotropic fluid, find different components of total energy per unit volume.
What are the consequences of Lorentz transformations on length and time when observed from a frame moving at relativistic velocities ?
Show that for very small velocity, the equation for kinetic energy, K=\Delta mc^2 becomes K=\dfrac{1}{2}m_0v^2, where notations have their usual meanings.
A particle P of mass m_1 collides with another particle Q of mass m_2 at rest. The particles P and Q travel at angles \theta and \phi, respectively, with respect to the initial direction of P. Derive the expression for the maximum value of \theta.
Two spaceships approach each other, both moving with same speed as measured by a stationary observer on the Earth. Their relative speed is 0.7c. Determine the velocity of each spaceship as measured by the stationary observer on the Earth.
Consider the diagram below with a water flow rate Q. Derive the expression for Q in terms of the difference in the manometer heights h and the cross-section areas A_1 and A_2:
Define moment of inertia and radius of gyration of a body of mass M rotating about an axis. State and prove Parallel Axis theorem on moment of inertia.
Consider two frames of reference S and S' having a common origin O. The frame S' is rotating with respect to the fixed frame S with a uniform \vec{\omega}=3a_x\,\mathrm{rad\,s^{-1}}. A projectile of unit mass at position vector \vec{r}=7a_x+4a_y\,\mathrm{m} is moving with \vec{v}=14a_y\,\mathrm{m\,s^{-1}}. Calculate in the rotating frame S' the following forces on the projectile:
(i) Euler's force
(ii) Coriolis force
(iii) Centrifugal force
A particle moving in a central force field describes the path r=ke^{\alpha\theta}, where k and \alpha are constants. If the mass of the particle is m, find the law of force.
(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.
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}.
