LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – November 2008
MT 5506 – MECHANICS – I
Date : 05-11-08 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 gT2=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 m1, m2, m3 respectively for which е is given are lying in a straight line. m1 is projected with a given velocity so as to impinge on m2 which in turn impinges on m3. If each impinging ball after impact is reduced to rest, prove that m22 = m1m3.
- Two particles of masses m1 and m2 (m1>m2) 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 m1 and m2 moving with velocities u1 and u2 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 1350 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 w1, w2 and μ1, μ2 respectively and μ1> tan α > μ2 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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