LOYOLA COLLEGE (AUTONOMOUS), CHENNAI-600 034.
M.Sc. DEGREE EXAMINATION – mathematics
FourTh SEMESTER – APRIL 2003
MT 4801/ M 1026 mechanies – II
16.04.2003
1.00 – 4.00 Max: 100 Marks
Answer ALL the questions
- a) Explain the term ‘ABERRATION’. Also derive the relativistic formula
for aberration in the form (8)
(OR)
- Show that the operator is an invariant for
Lorentz transformation. (8)
- a) State ‘ETHER’ Hypothesis. Explain the Michelson- Morley experiment
and give the conclusion. (17)
(OR)
- b) Show that Lorentz transformations forma group . (17)
- a) Obtain the transformation formula for mass in the form (8)
(OR)
- If a body of mass m disintegrates while at rest in to two parts of rest masses m1 and m2, show that the energies E1 and E2 of the parts are
given by (17)
- a) Derive the equation E = m C2, Deduce that p2 – is an invariant under
Lorentz transformation. (17)
(OR)
- Obtain the transformation formula for force components in the
(17)
- a) Explain ‘contravariant vectors’, covariant vectors,
‘contravariant tensors’ and ‘covariant tensors’. (8)
(OR)
- If a vector has components, on cartesion coordinates then
the components in polar coordinates are and if the components be
then the polar coordinates components are (8)
- a) Define fundamental tensors and show that gmg is a Covariant tensor of
rank two. Also transform ds2 = dx2 + dy2 + dz2 in polar and cylindrical
coordinates. (17)
(OR)
- b) Define Christoffel’s 3-index symbols of the first and second kind. Also
calculate christoffel’s symbols corresponding to the metric
ds2 = dr2 + r2dq2 + r2sin2q df2. (17)
- a) Define ‘Energy Tensor’. Show that the equationfor m = 4
gives the equation of continuity in Hydrodynamics. (8)
(OR)
- Obtain isotropic polar coordinates and Cartesian coordinates.
Also Deduce that the velocity of light at distance r1 from the origin is
(8)
- a) Obtain the schwarzchild line element in the neighbourhood of an attracting particle
in the from (17)
(OR)
- b) Derive the differential equation to the planetary orbits in the
form . (17)