LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008
MT 5506 – MECHANICS – I
Date : 051108 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer ALL the questions. (10 X 2 = 20)
 When two forces of equal magnitudes are inclined at the angle 2α, their resultant is twice as great as when they are inclined at an angle 2β. Prove that Cos α = 2 Cos β.
 State the triangle law of forces.
 State Newton’s laws of motion.
 Define angle of friction.
 If T is the time of flight, R the horizontal range and α, the angle of projection, show that gT^{2}=2R tan α.
 State Newton’s experimental law on impacts.
 Define the limiting velocity in a resisting medium.
 Define a couple and the moment of a couple.
 A particle is moving with uniform acceleration in a straight line, velocity u at A and V at B. Find the velocity at the midpoint of AB.
 Define relative angular velocity.
PART – B
Answer any FIVE questions only. (5 X 8 = 40)
 The angle between two forces of magnitudes P + Q and P – Q is 2α and the resultant of the forces makes an angle f with the bisection of the angle between the forces. Show that p tan f = Q tan α.
 State and prove Lami’s theorem.
 Discuss the motion of a particle moving along a straight line with uniform acceleration f.
 Two like parallel forces P and Q (P > Q) act at A and B respectively. If the magnitudes of the forces are interchanged, show that the point of application of the resultant on AB will be displaced through the distance.
 Three balls of masses m_{1}, m_{2}, m_{3} respectively for which е is given are lying in a straight line. m_{1} is projected with a given velocity so as to impinge on m_{2} which in turn impinges on m_{3}. If each impinging ball after impact is reduced to rest, prove that m_{2}^{2} = m_{1}m_{3}.
 Two particles of masses m_{1} and m_{2} (m_{1}>m_{2}) are connected by means of a light in extensible string that passes over a light, smooth, fixed pulley. Discuss the motion.
 Two smooth spheres of masses m_{1} and m_{2} moving with velocities u_{1} and u_{2} respectively in the direction of line of centres impinge directly. Discuss the motion of each mass after impact, given that e is the coefficient of restitution.
 Show that the velocity with which a particle must be projected down a smooth inclined plane of length and height h so that the time of decent shall be the same as taken by another particle in falling freely through a distance equal to the height of the plane is.
SECTION – C
Answer any Two questions. (2 X 20 = 40)
 (a) Three equal strings of no sensible weight are knotted together to form an equilateral and a weight W is suspended from A. If the triangle and the weight be supported with BC horizontally by means of two strings at B and C each at an angle 135^{0} with BC, show that the tension in BC is .
(b) A weight is supported on a smooth plane inclined at an angle α with the horizon, by a string inclined to the vertical at the angle β. If the inclination of the plane is increased to ٧ and the inclination of the string with the vertical is unaltered, the tension in the string is doubled in supporting the weight. Prove that (10+10)
 (a) State and prove Varignon’s theorem.
(b) Two rough particles connected by a light string rest on an inclined plane. If their weights and corresponding coefficients of friction are w_{1}, w_{2} and μ_{1}, μ_{2} respectively and μ_{1}> tan α > μ_{2 }where α is the inclination of the place with the horizon, prove that , if both particles are on the point of moving down the plane.
 (a) A particle is projected vertically upwards with the velocity of and after t seconds, another particle is projected upwards from the same point with the same velocity. Prove that the particles will meet at a height after a time seconds from rest.
(b) Discuss the motion of a particle falling under gravity in a medium whose resistance
varies as the square of the velocity.
 Show that the path of a projectile is a parabola. Also show that the speed of a projectile at any point on its path equals the speed of a particle acquired by it in falling from the directrix to that point.
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