LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – April 2009
ZA 25
MT 5500 - MECHANICS - II
Date & Time: 24/04/2009 / 1:00 - 4:00 Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions:
(10 x 2 = 20)
1. Write down the centre of gravity of a thin uniform triangular lamina.
2. Define centre of gravity of a compound body.
3. State the principle of virtual work.
4. Write down the intrinsic equation of a catenary.
5. Define sag and span of a catenary.
6. Define periodic time.
7. Define seconds pendulum.
8. Define central orbit.
9. Define compound pendulum.
10. Define radius of gyration.
PART – B
Answer any FIVE questions:
(5 x 8 = 40)
11. Find the centre of gravity of a uniform solid right circular cone.
12. A piece of uniform wire is bent in the shape of an isosceles triangle whose sides are a, a and
a 2a  b
.
2 2a  b
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 plane; at the middle point of the opposite rod a weight W1 is placed. If W be the
3W  W 
weight of each rod, show that tension in the string is
.
3
14. Find the resultant of two simple harmonic motions with same period in the same straight line.
15. A particle moves in a SHM in a straight line. In the first second, after starting from rest, it
travels a distance a and in the next second, it travels a distances b in the same direction.
b. Show that the distance of the centre of gravity from the base of the triangle is
2a 2
.
3a  b
16. Find the components of velocity and acceleration along the radial and transverse directions.
17. The velocity of a particle along and perpendicular to radius vector from a fixed origin are
1
1
 r 2 and  2 . Prove that the polar equation of the path is

 c and the components
2
2 r

Prove that the amplitude of the motion is
2
3
and  r  2
.
r
r
18. An elliptic lamina of semi axes a and b swings about a horizontal axis through one of the foci
in vertical plane. Find the length of SEP.
of the acceleration are 2 r  
2 3
2
2
1
2
PART – C
Answer any TWO questions:
(2 x 20 = 40)
19. (a) Find the centre of gravity of the area enclosed by the parabolas y 2  ax and
x 2  by (a  0, b  0) .
(b) Four equal rods, each of length a, are jointed to form a rhombus ABCD and the points B
and D and joined by a string of length . The system is placed in a vertical plane with A
resting on a horizontal plane and AC vertical. Prove that the tension in the string is
2Wl
where W is the weight of each rod.
4a 2  l 2
20. a) Find the equation of the catenary in the Cartesian and polar forms.
b) A telegraph wire, stretched between two points at a distance ‘a’ feet apart, sags ‘n’ feet in
 a 2 7n 
the middle. Prove that the tension at the ends is approximately w    where ‘w’ is the
 8n 6 
weight per unit length of the wire.
21. a) Find the moment of inertia of a thin uniform parabolic lamina bounded by the parabola
y 2  4a(h  x) and y axis about the Y axis.
b) The law of force is u 5 and a particle is projected from an apse at adistance ‘a’. Find the
orbit when the velocity of projection is

.
2a 2
22. a) Prove that the motion of a compound pendulum is the same as that of a simple equivalent
k2
.
h
b) A solid sphere rolls down an inclined plane of inclination  to the horizon which is
sufficiently rough to prevent sliding. Show that the acceleration is always constant.
pendulum of length
**************
2