Gauhati University 2007 Previous : thetics--IV - Question Paper

2007
MATHEMATICS
FOURTH PAPER
(Tensors and Mechanics)
Full Marks: 80
Time: three hours
The figures in the margin indicate full marks for the ques.
PART-A (Objective-type Questions)
(Marks: 32)
1. ans the subsequent questions: 1×10=10
(a) What is the relation ranging from contravariant and covariant vectors with reference to
rectangular coordinate transformation?
(b) Why gij in ds2=gijdxidxj is called the fundamental tensor?
(c) State the difference of gijAj and BijAj, if any.
(d) When can we shift the point of application of a force F to any point on its line of
action?
(e) A system of forces acting on a rigid body is equivalent to_____________. (Fill up
completely)
(f) If P and Q are 2 non-intersecting forces whose directions are perpendicular, the
distances of the central axis from their lines of action are as x: y; elaborate x and
y?
(g) At the clamped end of a beam,
State the physical condition.
(h) If A, B and C are the moments of inertia of a body about 3 mutually
perpendicular axes, then what is the condition so that C=A+B holds?
(i) What is the moment of inertia of a right circular cone about a slant height?
(j) Are the velocity components in cylindrical polar coordinates orthogonal? provide
reasons.
2. ans any 5 parts of the following: 2×5=10
(a) From the derivation principle of the legal regulations
discuss the cause of the use of the name ‘contravariant’.
= 0
dx
dy
a
a A
x
x
A
i
i
¶
¶ ¢ ¢ =
(b) Partial derivative of a tensor of rank 1 is not a tensor. provide an example that the
difference of 2 such derivatives is a tensor. Justify.
(c) Deduce the condition that a system of forces acting on a body be equivalent to a
single force.
(d) Derive the expressions of invariant volumes for a system of forces.
(e) A circular disc of radius r oscillates about an axis perpendicular to its plane at a
distance h from the centre. obtain the length of simple equivalent pendulum.
(f) Write down the general equations of motion of a rigid body and discuss their
individual importance.
(g) What do you mean by intrinsic formula of a path defined by a particle? Why is
it termed so?
(h) In a central orbit, defined by a particle, the velocity at any point is inversely
proportional to the distance of the point from the centre of force. obtain the legal regulations of
force.
3. ans any 4 parts of the following: 3×4=12
(a) Show that every second-order tensor Aij (or Aij) can be expressed as the sum of a
symmetric and an antisymmetric tensors.
(b) If ijk is the permutation tensor, show that 123123=3 in 3 dimensions.
(c) obtain the moment of inertia of a circular disc of radius
and of mass M about a
tangent.
(d) Show that the motion of a particle in a conservative field of force is independent
of the path.
(e) Show that the Central Axis for a system of forces is unique.
(f) Show that the resulting for a system of forces acting on a body is the identical
irrespective of referential base point.
PART-B (Subjective-type Questions)
(Marks: 48)
4. (a) Derive the xk – covariant derivative of the second-order contravariant tensor Aij
with respect to the fundamental tensor gij. 8
(b) Evaluate the non-vanishing Christoffel’s 2nd brackets for the metric 8
Or,
(i) Show that a vector of constant magnitude is orthogonal to its intrinsic derivative.4
(ii) Deduce the expression of Laplacian of a vector 4
5. (a) Any number of wrenches of the identical pitch p act along the generators of the identical
system of the hyperboloid
ds two = dr two + r 2dq two + r two sin 2q df 2
A
1 2
2
2
2
2
2
+ - =
c
z
b
y
a
x
Show that they will decrease to a single resulting given their central axis is
parallel to a generator of the cone 8
(b) Show that a provided system of forces can be changed by 2 forces can be changed by
two forces equivalent to the provided system in an infinite number of ways and the
tetrahedron formed by the 2 forces is of constant quantity. 8
Or,
A heavy rod AB rests on 3 horizontal supports A, B and C; C being midway ranging from
them and ABC horizontal. If the rod be slightly deflected, obtain its form. Hence show that
for maximum sag,
6. (a) A particle of mass m moves in a resisting medium under a central attraction mP.
Show that the formula of the orbit is
R is the resistance of the medium per unit of mass and h0 is the initial moment of
momentum about the centre of force. 8
Or,
What is the clear physical situation that admits inverse square law? explain the
dynamical significance of Kepler’s laws of planetary motion.
(b) A particle moves on a smooth sphere under no forces other than the pressure of the
surface. Show that the path of it is provided by the formula cot = cotcos where  and 
are its angular coordinates and  =  initially. 8
Or,
A circular area can turn freely about a horizontal axis which passes through a point O of
its circumference and is perpendicular to its plane. If the motion commences when the
diameter through O is vertically above O, show that when the diameter has turned
through an angle, the components of strain at O along and perpendicular to the diameter
are respectively
Where W is the weight of the circular area.
0 two 2 =




 - +




 + +




 +
c
ab
y p
b
ca
x p
a
bc
p
16
1 33
. = = a + x a and x
 + = = - dt
u
R
where h h e
h u
p
u
d
d u
2 two 2 0
2
,
q
( q ) sinq
3
7 cos 4
3
W
and
W -

Earning:   Approval pending.