LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – SUPPLEMENTARY – JUNE 2012
MT 6604/MT 5500 - MECHANICS - II
Date : 30-06-2012
Time : 2:00 - 5:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions:
(10 x 2 = 20 marks)
1. Define centre of gravity.
2. Write down the formula for centre of gravity of a uniform wire in the form of an arc of a plane curve.
3. Define Virtual work.
4. Define suspension bridge.
5. Define simple pendulum.
6. Define centripetal force.
7. Write down the components of velocity along the radial and transverse direction.
8. Write down the p – r equation of hyperbola.
9. Define moment of inertia.
10. Define simple equivalent pendulum.
PART – B
Answer any FIVE questions:
(5 x 8 = 40 marks)
11. Find the centre of gravity of a uniform solid right cirucular cone.
12. Find the centre of gravity of uniform circular arc subtending angle 2 at the centre.
13. A regular hexagon is composed of six equal heavy rods freely jointed together and two opposite
angles are connected by a string which is horizontal, one rod being in contact with a horizontal plang;
at the middle point of the opposite rod a weight W is placed. If W be the weight of each rod, show
3W  W
that the tension in the string is
.
3
x
14. Derive the equation of the common catenary in the form y = v cosh .
c
15. A particle moves in a simple Harmonic Motion in a st. line. In the first second, after starting from
rest, it travels a distance ‘a’ and in the next second, it travels a distance ‘b’ in the same direction.
2a 2
Prove that the amplitude of the motion is
.
3a  b
16. Derive the differential equation of the central orbit in polar coordinates.
17. Find the moment of intertia of the square lamina about a diagonal of length  .
18. Prove that kinetic energy of a rigid body moving in two dimension is equal to the k. E due to
translation X k . E due to rotation.
PART – C
Answer any TWO questions
(2 x 20 = 40 marks)
19. a) A homogeneuous solid is formed of a hemisphere of radius r soldered to a right circular cylinder of
the same radius. If h be the height of the cylinder, show that the centre of gravity of the solid from
3 2h 2  r 2 
.
the common base is 
4 3h  2r 
b) Find the centre of gravity of the area enclosed by the parabolas y 2  ax and
x 2  by
(a 0, b  0).
(10 + 10)
20. a) Four equal rods, each of length a, are joined together to form a rhombus ABCD and the points B
and D are joined by a string of length  . The system is placed in a vertical plane with A resting on
2W
a horizontal plane and AC vertical. Prove that the tension in the string is
where W is the
4a 2   2
weight of each rod.
b) A uniform chain, of length  , is to be suspended from two points A and B, in the same horizontal line
so that either terminal tension is n time that at the lowest point. Show that the span AB must be
1
log n  n 2  1 .
(10 + 10)
2
n 1


21. a) Find the resultant of two simple harmonic motions of the same period in the same straight line.
b) A particle p describes the orbit r n  a n cos n  under a central force, the pole being the centre.
Find the law of force.
(10 + 10)
2
Mr
22. a) Show that the M.I of the parabolaid of revolution about its axis is
where M is the mass and
3
r i radius A its base.
b) A uniform square plate is allowed to make small oscillations in it its own plane about an axis
2 2
through a corner and perpendicular to the plane. Show that the length of the S.E.P is
3
times length of the side of the plate.
(10 + 10)
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