PYQ Bank

(a) How are linear and angular momenta related to each other ? Considering a system of mass points under the influence of forces derived from potentials only, prove that the generalised linear momentum is conserved. Establish further that the angular momentum of the above system is also conserved if the potential is centrally symmetric.

(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.

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.

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.

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.

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 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.

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?

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?

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.

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.

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?

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.

A system with 2 independent coordinates x_1 and x_2 has the following Larangian L = \frac{1}{2} \frac{(\dot{x}_1)^2}{\alpha + \beta x_2} + \frac{1}{2} (\dot{x}_2)^2 (x_2)^2 - \gamma x_2 \alpha, \beta, \gamma being constants. Obtain the Lagrangian equation of motion.

A star of the size of our sun having a radius of 7 \times 10^5\text{ km} and having a mass 2 times as that of the sun is rotating at a speed of one revolution every 10 days. If the star undergoes a gravitational collapse to a neutron star of radius 10\text{ kms}, assuming that the star is a uniform sphere at all times, calculate the rotation speed of the neutron star. The moment of inertia of solid about its diameter is 2/5\text{ MR}^2 where M and R are the mass and radius of the solid sphere respectively.