(b) What are constraints of motion ? Explain with examples the holonomic and non-holonomic constraints. Discuss critically how can one overcome the limits of constraints by introducing generalised coordinates.
What do you mean by the moments and products of inertia ? Show that the angular momentum vector is related to the angular velocity components by linear transformation relations.
Write down the expression for the relativistic mass of a particle moving with a velocity v in terms of its rest mass. Establish from the above expression Einstein's mass-energy relation E = mc^2.
A particle moves in space with the Lagrangian L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - V + A \dot{x} + B \dot{y} + C \dot{z} where A, B, C are given functions of x, y, z. Find the corresponding Hamiltonian in terms of coordinates and momenta.
For a freely falling body from the height h' on the surface of the earth in the northern hemisphere with a latitude \theta, show that the deviation of the body towards east at the final stage is given by \frac{1}{3} \omega \cos \theta \left(\frac{8h^3}{g}\right)^{1/2}, where \omega is the angular velocity of the earth and 'g' is the acceleration due to gravity.
If a single stage rocket is fired vertically from rest at the earth's surface burns its fuel in a time of 30 sec and their relative velocity = 3 km sec^{-1}, what must be the mass ratio m_0/m for a final velocity v of 8 km sec^{-1} ?
A meson of rest mass \pi comes to rest and disintegrates to a muon of rest mass \mu and a neutrino of zero rest mass. Show that the kinetic energy of motion of the muon is T = \frac{(\pi - \mu)^2 c^2}{2 \pi}
Write the Lagrange's equation for a system of particles which is acted upon by Conservative forces. What is a cyclic coordinate ? Show that the generalized momentum conjugate to a cyclic coordinate is conserved.
An observer S_1 sees two bodies A and B having equal rest mass approach each other with equal but opposite velocities 4c/5. To a second observe S_2, the body A is at rest. What us the velocity of the body B as seen by observe S_2? What are the kinetic energies of the body B in the frames of S_1 and S_2?
Write the Euler's equations for the rotational motion of a rigid body with one point fixed, under the action of torque N. Apply these equations to discuss the rotational motion of a symmetrical top in the absence of any force other than the reaction at the fixed point.
A block of mass m, attached to an ideal massless spring of force constant k, is at rest on a smooth horizontal floor. A second block of mass 2 m, moving with an initial velocity strikes the free end of the spring. The collision is one dimensional and elastic. (i) Calculate the maximum compression of the spring. (ii) What are the velocities of the block long time after the collision?
Derive the relationship between the impact parameter and the scattering angle for the scattering of an particle of charge +2e by a nucleus of charge +Ze. Calculate the impact parameter for an angle of deflection of 30^\circ if the kinetic energy of the alpha particle is 6 \times 10^{-13}\text{ J}.
Using the Lagrangian for the system of a planet and the Sun obtain the equation of motion. Use them to get the equations for the orbit.
Using the rocket equation and its integral find the final velocity of a single stage rocket. Given that
(i) the velocity of the escaping gas is 2500\text{ m/s},
(ii) the rate of loss of mass is. (where m_0) is the initial mass and 0.27 m_0 is the final mass).
Two spaceships are moving at a velocity of 0.9\text{ c} relative to the Earth in opposite directions. What is the speed of one spaceship relative to the other? (c = \text{velocity of light})
An observer A sees two events at the same space point (\Delta x = \Delta y = \Delta z = 0) and separated in time by \Delta t = 10^{-6}\text{ s}. Another observer B sees them to be separated by \Delta t' = 3 \times 10^{-6}\text{ s}. What is the separation in space of the two events as observed by B? What is the speed of B relative to A?
A person in a spaceship is holding a rod of length of 0.5 m, the space ship is cruising at a speed V parallel to the earths surface. What does the person in the space ship notice as the rod is rotated from parallel to perpendicular to the space ships motion? What does an observer on earth's surface notice?
Determine equation of trajectory for a particle under central force `F', the magnitude of which is given by F = -\frac{A}{r^2} + \frac{B}{r^3} where A and B are positive constants.
The mass of muon at rest is 207\text{ m}_e where m_e is the electron rest mass (0.511\text{ MeV}). The mean life time at rest for muon is 2.2\ \mu\text{s}. The life time of muon emerging from an accelerator is measured in the laboratory as 6.9\ \mu\text{s}. Estimate the speed of these muons in the laboratory.