Aligarh Muslim University (AMU) 2010 B.Sc Physics Classical mechanics & SPLRelativity - Question Paper

(4293)

2009-2010

B.Sc. (HONS.) (PART-III) EXAMINATION (PHYSICS)

CLASSICAL MECHANICS & SPL. RELATIVITY (PH - 308)

Maximum Marks : 40    Duration : Three Hours

NOTE: (i) Answer the following questions.

(ii)    Symbols/notations, wherever not explained, have their usual meaning.

(iii)    Use of calculator is permitted.

1.    (a) Explain, giving appropriate examples, what are holonomic constraints. Explain what (03)

difficulties do the constraints of motion introduce in the solution of mechanical problems and how are these difficulties removed.

(b) Express the kinetic energy of a system of particles in terms of generalized coordinates. (04) What will be its nature if the transformation equations defining generalized coordinates do not contain the time explicitly ?

OR

l⁷. (a) Define the terms : generalized coordinates and generalized momentum. Derive the (03) relation which represents DAlemberts principle.

(b)    Show that : - = L    (01)

dq_} dq_j

(c)    Construct the Lagrangian and hence obtain the equation of motion of a simple (03) pendulum.

2.    (a) State and explain Hamiltons principle and using this principle derive Lagranges (04)

equations of motion.

(b) Show that the minimum distance between two points in a plane is a straight line.    (02)

3.    (a) Set up the Lagrangian and equations of motion of a particle moving under the influence (05)

of a central force. Hence obtain the differential equationof its orbit.

(b) Obtain the relation which represents virial theorem. Use this theorem to derive Boyles (02) law for perfect gases.

OR

3¹. (a_ Derive the equation of the orbit of a particle moving under the influence of an attractive (05) inverse .square law force. What will be the shape of the orbit for : (i) E > 0 and (ii)

E<0? *

(b) The orbit described by a particle under a central force is given by r = a (l+cos0), where (02) a is a constant. Find the force law.

4.    (a) Derive the Rutherford scattering cross-section. Explain why does the total cross-section (05)

tend to become infinite.

COHtd $ 9**2

a

(b) A proton collides elasticity with a pion at rest in Lab. frame. Calculate the maximum scattering angle in the Lab. frame, (0i)max-

5.    Answer any THREE parts of the following :

(a)    Explain the inner and outer products of two tensors.

(b)    Write the Newtons equation of motion in covariant form. Discuss the space and time parts of this equation.

Jc) Calculate the threshold energy of the incident proton for the following reaction :

p + p-> n⁰ + p n + 7T⁺

(Take m_p = m_n = 940 Mev/c and m_K+ = m_K0 ⁼ 140 Mev/c )

(d)    Write the formulae for the kinetic energy and momentum of a relativistic particle in terms of its Lorentz factor. Calculate the values of these quantities for an electron of rest

mass 0.51 MeV/c² moving with a velocity {y/2/3c.

(e)    Derive expressions for Lagrangian and Hamiltonian of a relativistic free particle.

6.    (a) Define the electromagnetic field tensor in terms of four-potential vector. Obtain the

elements of in terms of the components of Ii and vectors.

OR

(a State and explain gauge transformation and gauge invariance of fields.

(b) Show that F= a_a a_vp F_ap and hence obtain the transformations for the components of E and B vectors.

Attachment: aligarh-muslim-university--amu--2010-b-sc-physics-264-1.pdf

Earning:   Approval pending.