LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 11

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)

1. 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.

2. State the triangle law of forces.
3. State Newton’s laws of motion.
4. Define angle of friction.

1. If T is the time of flight, R the horizontal range and α, the angle of projection, show that gT²=2R tan α.
2. State Newton’s experimental law on impacts.
3. Define the limiting velocity in a resisting medium.
4. Define a couple and the moment of a couple.
5. 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.
6. Define relative angular velocity.

PART – B

Answer any FIVE questions only.                                                         (5 X 8 = 40)

1. 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.
2. State and prove Lami’s theorem.
3. Discuss the motion of a particle moving along a straight line with uniform acceleration f.
4. 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.
5. 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.
6. 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.
7. 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.

 

 

1. 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)

19. (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)

20. (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.

21. (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.

22. 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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