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
01. a) Explain the term ‘ABERRATION’. Also derive the relativistic formula
for aberration in the form
tan  ' 
sin 


  cos   
c

where  
1
1
(8)
2
c2
(OR)
2
2
2
1 2


b) Show that the operator   2 
is an invariant for
x
 y 2  z 2 c 2 t 2
Lorentz transformation.
(8)
02. a) State ‘ETHER’ Hypothesis. Explain the Michelson Morley experiment
and give the conclusion.
(17)
(OR)
b) Show that Lorentz transformations forma group
.
(17)
 v 2u x
1  2
C
03. a) Obtain the transformation formula for mass in the form m'  m 
v2
1 2
C



(8)
(OR)
b)
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
C2 2
C2 2
m  m12  m22 and E 2 
m  m12  m22
given by E1 
(17)
2m
2m

04. a)

Derive the equation E = m C2, Deduce that p2 
Lorentz transformation.


E2
is an invariant under
C2
(17)
1
(OR)
b) Obtain the transformation formula for force components in the
v


C2
form F x  Fx 
 u y Fy  u z Fz 
 vux  

1  2 
C 

(17)
v2
V2
F
1

z
C2
C2
F y 
and F z 
 vux 
 vux 
1  2 
1  2 
C 
C 


Fy 1 
05. a)
Explain ‘contravariant vectors’, covariant vectors,
‘contravariant tensors’ and ‘covariant tensors’.
(8)
(OR)
b) If a vector has components x , y on cartesion coordinates then
the components in polar coordinates are r, and if the components be
2r
x , y then the polar coordinates components are r  r 2 , 
(8)
r
06.
a)
b)
07.
a)
Define fundamental tensors and show that g is a Covariant tensor of
rank two. Also transform ds2 = dx2 + dy2 + dz2 in polar and cylindrical
coordinates.
(17)
(OR)
Define Christoffel’s 3index symbols of the first and second kind. Also
calculate christoffel’s symbols corresponding to the metric
ds2 = dr2 + r2d2 + r2sin2 d2.
(17)
Define ‘Energy Tensor’. Show that the equation
 T 
 O for  = 4
 xr
gives the equation of continuity in Hydrodynamics.
(8)
2
b)
(OR)
Obtain isotropic polar coordinates and Cartesian coordinates.
Also Deduce that the velocity of light at distance r1 from the origin is

m
1 

 2r1 

m
1 

 2r1 
2 1
(8)
08. a) Obtain the schwarzchild line element in the neighbourhood of an attracting particle
in the from ds 2  
b)
 2m  2
dr 2  r 2 d  2  r 2 sin 2  d 2  1 
dt
r


2
m


1 

r 

1
(17)
(OR)
Derive the differential equation to the planetary orbits in the
d2 u
m
d
form
 u  2  3mu 2 where r 2
 h.
2
ds
d
h
(17)
**** *
3