MA/MSCMT-05
June Examination 2016
M.A. MSc. (Previous) Mathematics Examination
Mechanics
Paper MA/MSCMT-05
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C. Write answer as per the given instructions.
Section A 8 × 2 16
(Very Short Answer Questions)
Note: Section contain 08 very short Answer type Questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit may be thirty words.
State Alembert's Principle.
In polar coordinates when the particle is moving along a circle of radius then give expressions for redial and transverse accelerations.
Write expression for K.E. of a rigid body in a two dimensional motion under finite forces.
Write Euler's equations of motion.
Define invariable line.
What do you mean by conservation forces?
Write the equations of motion of a top.
(viii) Define stream function.
Section B 4 × 8 32
(Short Answer Questions)
Note: Section contain 08 short answer type questions.
Examinees will have to answer any four questions.
Each question is of 08 marks. Examinees have to delimit
each answer in maximum 200 words.
State and prove Alembert's Principle.
A uniform thin circular disc is set rotating with an angular
velocity w about an axis through the centre making an angle i
with the normal. Prove that the semi-vertical angle θ of the core
described by the axis of disc is given by tan tan θ 2
1 tani.
A small insect moves along a uniform bar of mass equal to itself
and of length 2a, the ends of which are constrained to remain on
the circumference of a fixed circle whose radius is a
3
2 . If the
insect starts from the middle point of the bar and move along
the bar with relative velocity show that the bar in time t will
turn through an angle tan
a
Vt
3
1 .
Drive the equation of motion of a simple pendulum by using
Lagrange's equations.
The velocity components for a two dimensional
flow system can be given in the Eulerian system by
u 2x 2y 3t, v x y t
2
1 Find the displacement of a
fluid particle in the Lagrangian system.
MA/MSCMT-05 2900 4 (P.T.O.)
470
Derive the equation of continuity by Euler's method.
State and prove Bernoulli's theorem.
Show that the velocity potential
log
x a y
x a y
2
1
2 2
2 2





h
h
gives a possible motion. Determine the stream lines.
Section C 2 × 16 32
(Long Answer Questions)
Note: Section contain 04 long answer type questions.
Examinees will have to answer any two questions. Each
question is of 16 marks. Examinees have to delimit each
answer in maximum 500 words. Use of non-programmable
scientific calculator is allowed in this paper.
10) Define the doublet and derive the complex potential for a
doublet. Also derive the image of a doublet with respect to a
circle.
11) Write down the condition for a surface representing a boundary
surface. Show that the ellipsoid
a k t
x
kt
b
y
c
z
1 2 2 2
2
2
2
2
2
n
n c
is a possible form of the boundary surface of a liquid.
MA/MSCMT-05 2900 4
470
12) A rectangular parallelepiped whose edges are 2a, 3a can
turn freely about its centre and is set rotating about a line
perpendicular to the mean axis and making an angle cos-1 8
5
with the least axis. Prove that ultimately the body will rotate
mean axis.
13) A uniform rod is placed with one end in contact with a horizontal
table, and is then at an inclination α to the horizon and is allowed
to fall. When it becomes horizontal, show that its angular
velocity is sin
3 a whether the plane be perfectly smooth
or perfectly rough. Show also that the end of the rod will not
leave the plane in either case.