LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 30

FIFTH SEMESTER – April 2009

MT 5506 – MECHANICS – I

 

 

 

Date & Time: 24/04/2009 / 1:00 – 4:00  Dept. No.                                                   Max. : 100 Marks

 

 

PART – A

Answer ALL the questions                                                                             (10 x 2 = 20 marks)

 

1. Define coplanar forces.
2. State the law of parallelogram of forces.
3. Define couple.
4. State the theorem on “equilibrium of three coplanar forces”.
5. State the principle of conservation of linear momentum.
6. Define angular velocity.
7. State Newton’s laws of motion.
8. Define horizontal range of a projectile.
9. Define impulsive force.
10. Define the coefficient of elasticity.

PART – B

Answer any FIVE questions                                                                     (5 x 8 = 40 marks)

 

1. State and prove Lami’s theorem.
2. A weight bar hangs by a string and is drawn aside by a horizontal force until the strings makes an angle 60^o with the vertical.  Find the horizontal force and the tension in the string.
3. 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 p w and q w are suspended from P and Q respectively, show that, when the reactions at R and S are equal the distance PR is given by .
4. 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 _, where α is the inclination of the plane with the horizon, prove that , if both particles are on the point of moving down the plane.
5. A and B describe concentric circles of radii ‘a’ and ‘b’ with speeds u and v, the motion being the same way round.  If the angular velocity of either with respect to the other is zero, prove that the line joining them subtends at the centre an angle whose cosine is .
6. Show that when masses P and Q are connected by a string over the edge of a table, the tension is the same whether P hangs and Q is on the table or Q hangs and P is on the table.
7. A particle is projected so as to graze the tops of 2 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 point reached in the flight.
8. 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 questions                                                                           (2 x 20 = 40 marks)

 

1. a) A uniform plane lamina in the form of a rhombus, one of whose angles is 120^o 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 is P>Q, then P² = 3Q².

1. b) Three equal strings of no sensible weight are knotted together to form an equilateral ∆ ABC and a weight W is suspended from A. If the triangle and the weight be supported with BC horizontal, by means of two strings at B and C each at an angle 135^o with BC, show that the tension in BC is .
2. a) State and prove Varignon’s theorem on moments.
3. b) A mass ‘m₁’ hanging vertically pulls a mass ‘m₂’ vertically up a rough inclined plane of inclination α by a light inextensible string passing over a smooth, light pulley at the top of the plane, the portion of the string between m₂ and the pulley being parallel to the line of greatest slope. Find the acceleration of the system and the tension in the string.

 

21. a) Two particles of masses m₁ and m₂ (m₁ > m₂) are connected by means of a light inextensible string passing over a light, smooth, fixed pulley. Discuss the motion.
22. b) A particle of mass m is projected vertically under gravity, the resistance of air being ‘mk’ times the velocity. Show that the greatest height altained by the particle is where V is the terminal velocity of the particle and λV is its initial velocity.

 

22. a) A particle is projected up an inclined plane of inclination b. Find the range on the inclined plane.
23. b) If V₁, and V₂ be the velocities at the ends of a focal chord of a projectile’s path and u, the horizontal component of the velocity, show that .

 

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