LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFOURTH SEMESTER – APRIL 2008MT 4501 – MECHANICS – I
Date : 29/04/2008 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00                                               PART – A Answer ALL questions.: (10 x 2 = 20)1. Define coplanar forces.2. Distinguish between internal and external forces.3. Define a couple.4. Define moment or torque of a force  .5. Define dynamical friction.6. Define angular velocity.7. A body of mass ‘m’ is carried in a lift moving with downwards acceleration f. Find the pressure on the lift.8. Define cone of friction.9. Define angle of projection.10. Define elastic body.
PART – BAnswer any FIVE  questions. (5 x 8 = 40)11. Two forces acting on a particle are such that if the direction of one of them is reversed, the direction of the resultant is turned through a right angle. Prove that the forces must be equal in magnititude.12. Prove that the sum of any two coplanar forces about any point in the plane of forces equals the moment of the resultant about that point.13. 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’ a part. If two 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  14. A system of forces in the plane of   is equivalent to a single force at A1 acting along the internal bisector of the angle BAC and a couple of moment G1. If the moments of the system about B and C are respectively G2 and G3, prove that  .15. 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 w.r.t the other is zero, prove that the line joining them subtends at the centre an angle whose cosine is  .16. Two particles of masses m1 and m2 (m1>m2) are connected by means of light inextensible string passing over a light, smooth, fixed pulley. Discuss the motion.17. If t be the time in which a projectile reaches a point P in its path and t1, the time from P till it reaches the horizontal plane through the point of projection, show that the height of P above the horizontal plane is  gtt1.18. 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 as (1+e): (1-e).
PART – CAnswer any TWO   questions. (2 x 20 = 40)19. a) State and prove Lami’s theorem.b) 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 horizontal by means of two strings at B and C each at an angle 135o with BC, show that the tension in BC is  .      (10+10)
20. a) A ladder which stands on a horizontal ground leaning against a vertical wall is so loaded that its centre of gravity is at the distances a and b from the lower and upper ends respectively. Show that if the ladder is in limiting equilibrium, its indination   to the horizontal is given by   where  and  1 are the coefficients of friction between the ladder and the ground and the wall respectively.b) A particle is projected upwards under the action of gravity in a resisting medium where the resistance varies as the square of the velocity. Discuss the motion.        (10+10)
21. a) Obtain the equation of the path of a projectile in Cartesian form.b) A particle is projected so as to clear two walls, first of height a at a distance b from the point of projection and the second of height b at a distance a from the point of projection. Show that the range on the horizontal plane is  and the angle of projection exceeds  .   (10+10)
22. a) Two smooth spheres 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.b) A body, sliding down a smooth inclined plane, is observed to cover equal distances, each equal to a, in consecutive intervals of time t1 and t2. Show that the indination of plane to the horizon is  .                  (10+10)

 

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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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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 15

FOURTH SEMESTER – April 2009

MT 4501 – MECHANICS – I

 

 

 

Date & Time: 02/05/2009 / 9:00 – 12:00 Dept. No.                                                   Max. : 100 Marks

 

 

 

PART – A

Answer ALL the questions:                                                                            10 x 2 = 20     

 

1. When do you say a concurrent system of forces is in equilibrium?
2. State the converse of Lami’s theorem.
3. Define moment of a force.
4. When do you say two unlike parallel forces form a couple?
5. Write down the components of the acceleration of a particle in the tangential and normal directions.
6. A ship is steaming north at and a man walks across its deek in a direction due west at 8 k.m.p.h. Find its resultant velocity in space.
7. State the principle of conservation of momentum.
8. Define coefficient of restitution.
9. Define (i) trajectory and (ii) horizontal range
10. Define limiting velocity.

 

PART – B

Answer any FIVE questions:                                                             5 x 8 = 40

 

11. State and prove Lami’s theorem.
12. A weight W hangs by string and is drawn aside by horizontal force until the string makes an angles 60^o with the vertical. Find the horizontal force and the tension in the string.
13. Find the resultant of two unlike parallel forces.
14. State the laws of friction.
15. A ship sails north-east at 15 kmph and to a passenger on board, the wind appears to blow from north with velocity of Find the true velocity of the wind.
16. A particle is dropped from the top of a tower and describes during the last second of its fall (9/25) of the height of the tower. Find the height of the tower.
17. A particle projected from the top of a wall 50 m. high, at an angle of 30^o above the horizon, strikes the level ground through the foot of the wall at an angle of 45^o. Show that the angle of depression of the point of striking the ground from the point of projection is .

 

 

 

 

18. A particle is projected from a point in a smooth fixed horizontal plane with velocity is at an elevation a. Show that the particle ceases to rebound from the plane at the end of time and that the total horizontal distance described in this period is .

PART – C

Answer any TWO questions:                                                              2 x 20 = 40

 

19. a) Three equal strings of no sensible wei0ght 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 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 .
20. b) The top of a pole is held by means of four horizontal wires which enert the following tensions: 20 lbs weight due North, 30 lbs weight due East, 40 lbs weight due South West and 50 lbs weight due South East. Find the magnitude and direction of the resultant pull on the post.
21. a) Discuss the equilibrium of a particle on a rough inclined plane acted on by an external force.
22. b) A uniform ladder rests at the angle 45^o with its upper extremity against a rough vertical wall and its lower extremity on the ground. If be the coefficients of friction between the ladder and the ground and the wall respectively, show that the least horizontal force which will move the lower extremity towards the wall is where W is the weight of the ladder.
23. Obtain the equation of the path of a projectile in Cartesian form. Also determine the time of flight, greatest height and horizontal range.
24. Two smooth spheres of masses m₁ and m₂ moving with velocities u₁ and u₂ imping directly. Obtain
25. the motion after impact
26. the impulse imparted to each other due to impact

• the change is kinetic energy due to impact.

 

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  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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

MT 5506/MT 4501 – MECHANICS – I

 

 

 

Date : 18-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL Questions                                                                                           (10 x 2 = 20 marks)

 

1. Define parallelogram of forces.
2. What are coplanar forces?
3. State the theorem on polygon of forces.
4. Define moment of a force.
5. State newton’s laws of motion.
6. Define angle of friction.
7. Define relative angular velocity.
8. State the principle of conservation of linear momentum.
9. Write down the “horizontal range” for projectile (with usual notations).
10. Define Newton’s experimental laws on impact.

PART – B

 

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

 

1. State and prove Lami’s Theorem.
2. A weight W hangs by a string and is drawn aside by a horizontal force until the string makes an angle 60^o with the vertical.  Find the horizontal force and tension in the string.
3. Find the resultant of two like parallel forces.
4. A uniform rod AB of length 2a and weight W is resting on two pegs C and D in the same level at a distance d apart.  The greatest weights that can be placed at A and B without tilting the rod are W₁ and W₂ respectively.  Show that .
5. A lift a ascends with constant acceleration f, then with constant velocity and finally stops under constant retardation f.  If the total height ascended is h and total time occupied is t, show that time during which the lift is ascending with constant velocity 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 projected upwards under the action of gravity in a resisting medium where the resistance varies as the square of the velocity.  Discuss the motion.
8. Two perfectly elastic smooth spheres of masses m and 3 m are moving with equal moments in the same st.line and in the same direction.  Show that the smaller sphere reduced to rest after it strikes the other.

PART – C

 

Answer any TWO questions                                                                                      (2 x 20 = 40 marks)

 

1. a) Three equal strings of no sensible weights are knotted together to form an equilateral triangle

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 .                                                                                           (10)

1. b) 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  . AB.                                                                    (10)

20. a) A system of forces in the plane of ABC is equivalent to a single force at A; acting along the

internal bisector of the angle BAC and a couple of moment G.  If the moments of the system

about B and C are respectively G₂ and G₃ prove that (b+c) G₁ = b G₂ + c G₂.

 

1. b) A ladder which stands on a horizontal ground leaning against a vertical wall is so loaded that

its centre of gravity is at the distance a and b from the lower and the upper ends respectively.

Show that if the ladder is in limiting equilibrium, its inclination  to the horizontal is given by

where m and m¹ are the coefficients of friction between the ladder and the

ground and the wall respectively.

 

21. a) A body, sliding down a smooth inclined plane is observed to cover equal distances, each equal

to a, in consecutive intervals of time t₁, t₂.  Show that the inclination of plane to the horizon is

.                                                                                                             (8)

1. b) Discuss the motion of two particles connected by a string.                                                 (12)
2. a) 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)

 

1. b) Show that the path of projectile in a parabola.                                                                     (10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

MT 5506/MT 4501 – MECHANICS – I

 

 

 

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

ANSWER ALL QUESTIONS:                                                                                (10 X 2 = 20 marks)

 

1. State the law of parallelogram of forces.
2. What is the magnitude and direction of the resultant of two forces and when and  are equal in magnitude and the angle between them is 60°?
3. Define torque of a force.
4. State any two laws of friction.
5. Define angular velocity.
6. State the triangle law of velocities.
7. State the principal of conservation of linear momentum.
8. A body of mass 50 gm is acted upon by a constant force F=100 dynes. Find the time required to move the body through a distance of 25 cms from rest.
9. Define time of flight of a projectile.
10. State Newton’s experimental law on impact.

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 120° is supported by two forces of magnitudes P and Q applied at the center in the directions of the diagonals so that one side is horizontal. Show that if P>Q, then P²=3Q
13. State and prove varignon’s theorem on moments.
14. Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the direction of P be reversed, the resultant is displaced through the distance .
15. A and B describe circles of radii ‘a’ and ‘b’ with speeds u and v, the motion being the same way round. If the angular velocity of either w.r.t. the other is zero, prove that the line joining them subtends at the center an angle whose cosine is
16. A train goes from one station to another moving during the first part of the journey with uniform acceleration ‘’ when the steam is shut off and brakes are applied, it moves with uniform retardation ‘’. if ‘a’ be the distance between the two stations, show that the time taken is .
17. 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 .
18. A ball impinges directly on 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 to that A is (1+e) : (1-e).

PART – C

 

ANSWER ANY TWO QUESTIONS:                                                               (2 X 20 = 40 marks)

 

19. a) Two weights P and Q are  suspended from a fixed point O by strings OA and OB and are kept apart by a

light rod AB. If the strings OA and OB make angle a and b with the rod, show that the angle q which

the  rod makes with the vertical is given by

1. b) 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 displaces though the

distance

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 tills. Show that weight of the rod is p+q and

that the centre of gravity of the rod divides AD in the ratio 2p+q : p+2q.

 

1. b) A uniform ladder rests at the angle 45° with its upper extremity against a rough vertical wall and its

lower extremity on the ground. If m and m¹ be the coefficients of friction between the ladder and the

ground and the wall respectively, show then the least extremity towards the wall is

where W is the weight of the ladder.

 

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 pully. Discuss the motion.

 

1. b) A particles of mass m is projected vertically under gravity, the resistance of air being ‘mk’ times

velocity.  Show that the greatest height attained by particle is where V is the

termind velocity of the particle and λV is its initial velocity.

22. a) Derive the equation of the projectile in the from .
23. b) Two smooth spheres of masses m₁ and m₂ moving with velocities u₁ and u₂ respectively in the direction

of the line of centers impinge directly. Discuss the motion.                                                         (10 + 10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

MT 5506/MT 4501 – MECHANICS – I

 

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

 

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

 

1. State the conditions for equilibrium of a system of concurrent forces.
2. State the law of parallelogram of forces.
3. Define torque of a force.
4. State any two laws of friction.
5. Find the resultant of two velocities 6 mt/sec and 8 mt/sec inclined to each other at an angle

of 30.

6. Define angular velocity.
7. Define momentum.
8. State the principle of conservation of linear momentum.
9. Define range of flight for a projectile.
10. Define the coefficient of elasticity.

 

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 120^° is supported by two forces of magnititudes 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 P² = 3Q².

 

13. State and prove Varignon’s theorem on moments.

 

14. Two particles weighing 2 kg and 1 kg are placed on the equally rough slopes of a double inclined plane whose inclinations with the horizontal are 60^° and 30^° The particles are connected by a light string passing over a smooth pulley at the common vertex of planes.  If the heavier particle is on the point of slipping downwards, show that the coefficient of friction is

 

15. A particle is dropped from an aeroplane which is rising with acceleration f and t secs after this; another stone is dropped. Prove that the distance of between the stones at time t^’ after the second stone is dropped is .
16. Two particles of masses m₁ and m₂ (m₁ > m₂) are connected by means of light inextensible string passing over a light, smooth, fixed pulley. Discuss the motion.

 

 

17. Show that when masses P and Q are connected by a string over the edge of a table, the tension is the same wheter P hangs and Q is on the table or Q hangs and P is on the table.

 

18. Two balls impinge directly and the interchange their velocities after impact. Show that they are perfectly elastic and of equal mass.

 

 

PART – C

 

Answer any TWO questions:                                                                 (2 x 20 = 40 marks)

 

19. a) Two strings AB and AC are knotted at A, where a weight W is attached. If the weight

hangs freely and in the position of equilibrium, with BC horizontal,

AB : BC : CA = 2 : 4 : 3, show that the tensions in the strings are

 

1. b) A system of forces in the plane of D ABC is equivalent to a single force at A, acting

along the internal bisector of the angle BAC and a couple of moment G₁.  If the moments

of the system about B and C are respectively G₂ and G₃,  prove that (b+c) G₁ = bG₂ + cG₃.

(10 + 10)

 

20. a) Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the

direction of P be reversed, the resultant is displaced through the distance .

 

1. b) A particle moving in a st. line is subject to a resistance KV³ producing retardation

where v is the velocity.  Show that if v is the velocity at any time t when the distance is

s,  and  where u is the initial velocity.                              (10 + 10)

 

21. Derive the equation to the path of the projectile in the form

 

22. A particle falls under gravity in a medium where the resistance varies as the square of the velocity. Discuss the motion.

 

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