LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc., DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – SUPPLEMENTARY – JUNE 2012
MT 5506/4501 – MECHANICS - I
Date : 27-06-2012
Time : 2:00 - 5:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions
( 10 x 2 = 20 marks)
1. What is the magnitude and direction of the resultant of two forces P and Q and
i)
When P and Q are equal in magnitude and angle between them is 60o?
2. State the conditions for equilibrium of a system of concurrent forces.
3. Define torque of a force.
4. State any two laws of friction.
5. Define angular velocity and angular acceleration.
6. Find the resultant of two velocities 6 mts/sec and 8 mts/sec inclined to each other at an angle of 30o.
7. Define momentum.
8. State the principle of conservation of linear momentum.
9. What is the time of flight of a projectile?
10. State Newton’s experimental laws.
PART – B
Answer any FIVE questions
(5 x 8 = 40 marks)
11. State and prove Lami’s theorem.
12. A uniform plane lamina in the form of a rhombus one of whose angles is 120o is supported by two
forces of magnitudes P and Q applied at the centre in the directions of the diagonals so that one side is
horizontal. Show that if P  Q , then P 2  3Q 2 .
13. State and prove Varignon’s theorem on moments.
14. A straight rod PQ of length ‘2a’ and weight W rests on smooth horizontal pegs R and S at the same
level at a distance ‘a’ apart. If two weights ‘pW’ and ‘qW’ are suspended from P and Q respectively,
show that when the reactions at R and S are equal, the distance PR is given by
a  1  3q  p 

.
2  1  q  p 
15. A particle is dropped from an aeroplane which is rising with acceleration ‘f’ and ‘t’ seconds afters
this, another stone is dropped. Prove that the distance between the stones at time ‘t’ after the second
1
stone is dropped is ( g  f ) t (t  2t ' ).
2
16. A particle falls under gravity in a medium where resistance varies as the velocity. Discuss the motion.
17. A particle is projected so as just to graze the tops of two walls, each of height 20 feet, at distances of
30 ft and 170 ft respectively, from the point of projection. Find the angle of projection and the highest
points reached in the flight.
18. A ball A impinges directly on an exactly equal and similar ball B lying on a smooth horizontal table.
If e is the coefficient of restitution, prove that after impact, the velocity of B is to that of A is
(1 + e) : (1-e).
PART – C
Answer any TWO question
(2 x 20 = 40 marks)
19. a) A light string of length  is fastened to two points A and B in the same level. A smooth ring of
weight W can slide on the string and the ring is in equilibrium vertically below B under the influence
aW
of a horizontal force of magnitude P. If AB = a, show that P 
and the tension in the string is

(Q 2  2 )
.
2 2
b) A square board ABCD of side a is fixed in a vertical plane with two of its sides horizontal. A string
of length ( 4a) passes over four smooth pegs at the angular points of the board and through a ring
of weight W which is below the lower horizontal side of the board. Prove that the tension in the string
W (  3a )
.
is
(10 + 10)
2  2  6 a   8a 2
W
20. a) A non uniform rod AD rests on two supports B and C at the same level where AB = BC = CD. If a
weight p is hung from A or a weight q is hung from D, the rod just tilts. Show that the weight of the rod
is p + q and that the centre of gravity of the rod divides A D in the ratio 2p + q : p + 2q.
b) A uniform ladder rests at the angle 45o with its upper extremity against a rough vertical wall and its
lower extremity on the ground. If  and 1 be the coefficients of friction of the ladder and the
ground between the wall respectively, show that the least horizontal force which will move the lower
W  1  2    1 
 where W is the weight of the ladder.
extremity towards the wall is 
(10 + 10)
2
1  1

21. a) Two particles of masses m1 and m2 (m1 > m2) are connected by means a light inextensible string
passing over a light, smooth, fixed pulley. Discuss the motion.
b) A lift ascends with constant acceleration f1 then with constant velocity and finally stops under constant
retardation f. If the total height ascended is h and the total time occupied is t1 show that the time during
4h
which the lift ascending with constant velocity is t 2  .
(10 + 10)
f
22. a) Derive the equation of the projectile in the form y  x tan  
gx2
.
2u 2 cos 2 
b) Two smooth spheres of masses m1 and m2 moving with velocities u1 and u2 respectively in the
direction of line of centres impinge directly. Discuss the motion.
(10 + 10)
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