LOYOLA COLLEGE  ( AUTONOMOUS ) , CHENNAI – 600 034

BSc DEGREE EXAMINATION  -MATHEMATICS

V SEMESTER – NOVEMBER 2006

 

Date       :                                                                                                                  Max  : 100 Marks

Duration:                                                                                                                  Hours: 3 hours

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SUB.CODE:MT5500                                                                                                                           SUB.NAME : MECHANICS-II

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Answer  ALL  the questions and each question carries 2 marks                     [  10 X 2  = 20  ]

 

01.State the cases of  non existence of center of gravity

02.State the forces which can be ignored in forming the equation of virtual work.

03.Define Neutral equilibrium with an example

04.Define Span of a Catenary

05.A particle is performing S.H.M. between points A and B. If the period of oscillation is

2p, show that the velocity at any point is a mean proportional between AP and BP.

06.Define Apse

07.If the angular velocity of a particle moving in a plane curve about a fixed origin is

constant, show that its transverse acceleration varies as radial velocity.

08.Find the M.I of a thin uniform rod.

09.Define radius of gyration.

10.State D’Alembert’s principle.

 

Answer any FIVE of the following                                                               [  5 X 8  = 40  ]

 

11. A uniform solid right circular cylinder of height l and base radius r is sharpened at

one end like pencil. If the height of the resulting conical part be h, find the distance

through which the C.G is displaced, it being assumed that there is no shortening of the

cylinder.

 

12.Find the C.G. of a uniform hollow right circular cone.

 

13.A uniform chain, of length l, is to be suspended from two points A and B, in the same

horizontal line so that either terminal tension is n times that at the lowest point. Show

that the span AB must be

14.A uniform string hangs under gravity and it is such that the weight of each element of

it is proportional to the projection of it on a horizontal line. To determine the shape of

the string.

 

15.Show that the composition of 2 simple harmonic motions of the same period along 2

perpendicular lines is an ellipse.

 

16.A particle executing S.H.M in a straight line has velocities 8,7,4 at three points distant

one foot each other. Find the period.

 

17.Derive the radial and transverse components of velocity and acceleration.

 

18. A circular disc of radius 5cms. Weighing 100 gms. is rotating about a tangent at the

rate of 6   turns per second. Find the frictional couple which will bring it to rest in one

minute

 

Answer any TWO of the following                                                                [  2 X 20  = 40  ]

 

19.i.Discuss the stability of a body rolling over a fixed body

ii.A body consisting of a cone and a hemisphere on the same base rests on a rough

horizontal table. Show that the greatest height of the cone so that the equilibrium may

be stable is  times the radius of the sphere.

 

20.i.State and prove the principle of virtual work for a system of coplanar forces acting on

a rigid body.

ii.A solid hemisphere is supported by a string fixed to a point on the rim and to a point

on a smooth vertical wall with which the curved surface of the hemisphere is in

contact. If  and are the inclination of the string and the plane base of the

hemisphere to the vertical, prove that

21.A point moves with uniform speed v along the curve r = a (1+ cosq ). Show that

1. Its angular velocity w about pole is
2. Radial component of acceleration is constant and equal to numerically

iii. Magnitude of resultant acceleration is

22.i.State and prove the theorem of parallel axes

1. Find the moment of inertia of a hollow sphere.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 18

FIFTH SEMESTER – APRIL 2008

MT 5500 – MECHANICS – II

 

 

 

Date : 28-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

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

 

1. Define centre of mass.
2. Define centre of gravity of a compound body.
3. State the principle of virtual work.
4. Define suspension bridge.
5. Define amplitude.
6. Define periodic time of S.H.M.
7. Write down the differential equation of the central orbit in p-r coordinates.
8. State the theorem of parallel axis.
9. Define equimomental system.
10. State D’Alemberts principle.

 

PART – B

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

 

1. A piece of uniform wire is bent in the shape of an isosceles triangle sides are ‘a’ ‘a’ and ‘b’. Show that the distance of C.G from the base of the triangle is .
2. A regular hexagon is composed of six equal heavy rods freely jointed together and two opposite angles are connected by string which is horizontal, one rod being in contact with a horizontal plane; at the middle point of the opposite rod a weight W’ is placed. If W be the weight of each rod, show that tension in the string is .
3. A particle moves in S.H.M in a straight line in the first second, after starting from rest, it travels a distance a and in the next second, it travels a distance b in the same direction. Prove that the amplitude of motion is .
4. The velocity of a particle along and perpendicular to radius vector from a fixed origin are and components of acceleration are and .
5. An elliptic lamina of semi axes a and b swings about a horizontal axis through one of the foci in a vertical plane. Find the length of the S.E.P.
6. Find the resultant of two simple harmonic motions of the same period in the same straight line.
7. Find the centre of gravity of a uniform solid right circular cone.
8. Find the components of the velocity and acceleration along radial and transverse directions.

 

 

 

PART – C

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

 

1. a) Find the centre of gravity of the area enclosed by the parabolas and .

1. b) A uniform chain, of length , is to be suspended from two points A and B, in the same horizontal line so that either terminal tension is n times that at the lowest point. Show that the span AB must be . …….(10+10)

1. a) Define catenary and derive the equation of the catenary.

1. b) Four equal rods, each of length a, are jointed to form a rhombus ABCD and the points B and D are joined by a string of length . The system is placed in a vertical plane with A resting on a horizontal pane and AC vertical. Prove that the tension in the string is where W is the weight of each rod.                 ……(10+10)

1. a) Obtain the differential equation of a central orbit in the form .

1. b) Show that the M.I about the x-axis of the parabola bounded by the latus rectum supposing the density at each point to vary as cube of the abcissa where M is the mass of the lamina.             ……(10+10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 18

FIFTH SEMESTER – November 2008

MT 5500 – MECHANICS – II

 

 

 

Date : 15-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

            PART-A

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

 

1. Define the centre of mass of a rigid body.
2. Where does the centre of gravity lie on a uniform solid tetrahedron?
3. State the principle of virtual work.
4. The maximum velocity of a particle moving in a simple harmonic motion is 2ft/sec and its period is Find its amplitude.
5. If the distance x of a point moving on a straight line from a fixed origin on it and its velocity V are connected by , show that the motion is simple harmonic.
6. Find the length of a seconds pendulum.
7. If the angular velocity of a particle moving in a plane curve about a fixed origin is constant, show that its transverse acceleration varies as radial velocity.
8. Define central orbit and central force.
9. State the perpendicular axes theorem.
10. State D’ Alemberts’ principle.

 

PART-B

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

 

11. Find the centre of gravity of a uniform solid hemisphere of radius .
12. A uniform circular lamina of radius 3a and centre O has a hole in the form of an equilateral triangle of side 2a with one vertex at O. Prove that the distance of the centre of gravity from

       O is   .

13. Derive the Cartesian equation of a catenary.
14. Show that the resultant of two simple harmonic motions of the same period in the same straight line is also a simple harmonic motion. Find the amplitude and epoch.
15. Obtain the radial and transverse components of velocity and acceleration in polar coordinates.
16. Find the law of force towards the pole when the central orbit is the curve .
17. If ℓ₁ be the length of an imperfectly adjusted seconds pendulum which gains n seconds per hour and ℓ₂ , the length of one which loses n seconds per hour in the same place, show that the true length of the seconds pendulum is    .
18. Find the moment of inertia of a truncated cone about its axis, radius of its ends being a and b.

 

 

PART-C

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

 

 

1. a) Four equal rods, each of length a, are joined to form a rhombus ABCD and the points

B and D are joined by a string of length ℓ. The system is placed in a vertical plane with A resting on a horizontal plane and AC vertical. Prove that the tension in the string is  where is the weight of the rod.

1. b) Find the centre of gravity of the area enclosed by the parabolas and  .                                                                                  (10+10)

 

1. a) A point moves with uniform speed v along the cardioids  Show that

1. its angular velocity  about pole is
2. radial component of acceleration is constant,

• magnitude of resultant is .

1. b) Derive the pedal equation of the central orbit.                                     (10+10)

 

1.  a) State and prove the parallel axes theorem concerning moment of inertia.

 

1. b) A solid sphere is rolling down a plane, inclined to the horizon at an angle α and rough enough to prevent any sliding. Find its acceleration.                         (10+10)

 

1. a)  A particle is performing simple harmonic motion of period T about a centre O and it

passes through the position P (where op = b) with velocity v in the direction op. Prove that the time that elapses before it returns to P is  .

 

1. b) Two particles of masses M and M¹ respectively are attached to the lower end of an elastic

       string whose upper end is fixed and are hung at rest. M¹ falls off. Show that the distance of    

       M from the upper end of the string at time t is   where a is the unstretched   

       length of the string; b and c are the distance by which it would be stretched when

        supporting M and M¹, respectively.                                                                                (10+10)

 

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 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 37

SIXTH SEMESTER – April 2009

MT 6604 – MECHANICS – II

 

 

 

Date & Time: 21/04/2009 / 9:00 – 12:00 Dept. No.                                                Max. : 100 Marks

 

 

PART – A

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

 

1. Define centre of mass.
2. Write down the formula for the C.G of a rigid body.
3. State the principle of virtual work.
4. Define suspension bridges.
5. Define Simple Harmonic motion.
6. Define second pendulum.
7. Write down the components of the acceleration of a particle in polar coordinates.
8. Define central orbit.
9. State the theorem of perpendicular axes.
10. State D’Alembert’s principle.

PART – B

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

 

11. A piece of uniform wire is bent in the shape of an isosceles triangle whose sides are a, a and b. Show that the distance of the C.G from the base of the triangle is .
12. Find the C.G of a uniform solid right circular cone.
13. A regular hexagon is composed of six equal heavy rods freely jointed together and two opposite angles are connected by a string which is horizontal, one rod being in contact with a horizontal plane; at the middle point of the opposite rod a weight W’ is placed. If W be the weight of each rod, show that tension in the string is .
14. Find the equation of the catenary in the Cartesian farm.
15. A particle moves in SHM in a straight line. In the first second, after starting from rest, it travels a distance ‘a’ and in the next second, it travels a distance ‘b’ in the same direction. Prove that the amplitude of the motion is .
16. Determine the maximum speed with which a car can turn round a level curve of radius 100 meters without slipping given that the coefficient of friction between the tyres and the track is 0.3.
17. Derive the Pedal equation or p-r equation of a central orbit.
18. Find the M.I of a hollow sphere.

 

 

                                                                   

                                                                 

PART – C

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

 

19. (a) Find the C.G of the area enclosed by the parabolas .

(b) ABCDEF is a regular hexagon formed of light rods smoothly jointed at their ends with a

diagonal rod AD. Four equal forces ‘P’ act inwards at the middle points of the rods AB, CD, DE, FA  and at right angles to the respective sides. Find the stress in the diagonal AD and state whether it is a tension or a thrust.                                                  (10+10)

20. (a) A string of length 2l hangs over two small smooth pegs in the same horizontal level. Show

that if h is the sag in the middle, the length of either part of the string that hangs vertically is

.

(b) Find the resultant of two simple harmonic motions of the same period in the same straight

line.                                                                                                            (10+10)

21. (a) State and prove the theorem of parallel axis.

(b) Find the M.I of the square lamina about a diagonal of length l.   (10+10)

22. (a) A particle acted on by a central attractive force is projected with a velocity at an

angle of with its initial distance ‘a’ from the centre of force. Show that the path is the equiangular spiral .

(b) A square lamina of side 2a rotates in a vertical plane about a horizontal axis passing through

one of the vertices and perpendicular to its plane and a weight equal to that of lamina is placed at the opposite vertex. Find the length of S.E.P.                        (10+10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

MT 6604/MT 5500 – MECHANICS – II

 

 

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

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

 

1. What is the Centre of Gravity of a compound body?
2. Where does the C.G of a uniform hollow right circular cone lie?
3. Define virtual work.
4. What is common catenary?
5. Show that frequency is the reciprocal of the periodic time in a simple harmonic motion.

6. If the maximum velocity of a particle moving in a simple harmonic motion is

2ft/sec and its period is 1/5 sec, prove that the amplitude is  feet.

7. What is the p–r equation of a parabola and an ellipse?
8. What are the radial and transverse components of acceleration?
9. Define moment of inertia?
10. Explain the conservation of angular momentum.

 

PART –B

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

 

11. A homogenous solid is formed of a hemisphere of radius r soldered to a right circular cylinder of

the same radius. If h be the height of the cylinder, show that  the center of gravity of the solid from

the common base is .

12. Find the center of gravity of a uniform trapezium lamina.
13. A uniform rod AB of length 2a  with one end A against a smooth vertical  wall being supported by

a string of length 2l, attached to the other end of the rod  B and to a point C of the wall vertically

above A.   Show that if the rod rests  inclined to the wall at an angle q, then  cos² q =.

14. Derive the intrinsic equation of the common catenary.
15. A second pendulum is in a lift which is ascending with uniform acceleration . Find the number of seconds it will gain per hour. Calculate the loss if

the lift were descending with an acceleration of .

16. Show that the composition of two simple harmonic motions of the same period

along two perpendicular lines is an ellipse.

17. Prove that the areal velocity of a particle describing a central orbit is constant.

Also show that its linear velocity varies inversely as the perpendicular distance

from the centre upon the tangent at P.

18. Show that the Moment of inertia of a truncated cone about its axis, the radii of its

ends being a and b, (a<b) is .

 

PART –C

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

 

19. (a) Find the centre of gravity of the area in the first quadrant bounded by the co-

ordinate axes and the curve .

 

(b) AB and AC are two uniform rods of length 2a and 2b respectively. If

, prove that the distance from A of the Centre of gravity of two the

rods is                                                                              (10 + 10)

 

20. (a) Show that the length of a chain whose ends are tied together and hanging over

a circular pulley of radius a, so as to be in contact with two thirds of the

circumference of the pulley is a    .

 

(b) Derive the expression for velocity and acceleration of a particle moving on a

curve.                                                                                                                                (10 + 10)

 

21. (a) A particle P describes the orbit under a central force. Find the

law of force.

 

(b) The law of force is  and a particle is projected from an apse at a distance

Find the orbit when the velocity of projection is  .                                                (10 + 10)

 

22. (a) State and prove Parallel axis theorem.

 

(b) Find the lengths of the simple equivalent pendulum, for the following:

1. i) Circular wire ii) Circular disc. (10 + 10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2012

MT 6604/MT 5500 – MECHANICS – II

 

 

 

Date : 18-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

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

 

1. State the conditions for non-existence of centre of gravity.
2. Mention the differences between center of gravity and centre of mass.
3. Define the work done by the tension in an elastic string.
4. Define Suspension bridge.
5. A particle executing simple harmonic motion makes 100 complete oscillations per minute and its maximum speed is 15 ft/sec. What is the length of its path and maximum acceleration?
6. Define Centripetal force.
7. Define Central Orbit.
8. An insect crawl at a constant rate u along the spoke of a earth wheel of radius a starting from centre, the wheel moving with velocity v. Find the accelerations along and perpendicular to the spoke.
9. State Parallel axis theorem.
10. Write down the formula for Moment of Inertia of a solid sphere and hollow sphere.

 

PART –B

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

 

11. Find the Centre of gravity of a uniform circular angle.
12. Derive the intrinsic equation of the catenary.
13. A particle executing simple harmonic motion in a straight line has velocities 8,7,4 at three points distant one foot from each other. Find the period.
14. ABCD is a trapezium in which AB and CD are parallel and of lengths a and b. Prove that the distance of the centre of mass from AB is where h is the distance between

AB and CD.

15. A particle describes the orbit rⁿ=Acosnθ-Bsinnθ under a central force, the pole being the centre. Find the law of force.
16. Derive the p-r equation of a central orbit.
17. Find the moment of Inertia of a thin uniform parabolic lamina bounded by the parabola y²=4a(h–x) about the y-axis.
18. State and prove D’Alemberts Principle.

 

PART –C

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

 

19. a) Find the centre of gravity of a hollow hemisphere.
20. b) Find the centre of gravity of the arc of the cardiod r = a(1+cosθ) lying above the initial

line.                                                                                                                                                   (8+12)

20. State and prove Principle of virtual work for a system of coplanar forces acting on a body.

 

21. a) Find the resultant of two simple harmonic motions of the same period in the same

straight line.

1. b) The speed v of a particle moving along the x-axis is given by

Show that the motion is simple harmonic with centre at x = 4b and amplitude 2b.

Find the time from x = 5b to x = 6b.                                                                                      (12+8)

22. a) If the law of acceleration is and the particle is projected from an apse

at a distance c with velocity , prove that the equation of the orbit is

1. b) Find the moment of inertia of an elliptic lamina.            (10 + 10)

 

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – NOVEMBER 2012

MT 6604 / MT 5500 – MECHANICS – II

 

 

 

Date :5/11/2012                Dept. No.                                        Max. : 100 Marks

Time :1.00 – 4.00

 

PART – A

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

 

1. State the conditions for non-existence of centre of gravity.
2. Write down the co-ordinates of center of gravity for a solid cone.
3. Define catenary.
4. Define suspension bridge.
5. A particle executing simple harmonic motion makes 100 complete oscillations per minute and its maximum speed is 15 feet per sec. What is the length of its path and maximum acceleration?

6. Write down any two applications of simple harmonic motion.
7. A point P describes with a constant angular velocity about O the equiangular

spiral r = a e^θ. O being the pole of the spiral. Obtain the radial and transverse

acceleration of P.

8. Define central orbit.
9. Define Moment of Inertia of a particle about a straight line.
10. Find the work that must be done on a uniform flywheel of mass 50 lbs and radius

6^״  to increase its speed of rotation from 5 to 10 rotation per second.

 

PART –B

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

 

11. Find the center of gravity of a hollow right circular cone of height h.
12. Prove that if a dynamical system is in equilibrium, then the work done by the

applied forces in a virtual displacement is zero.

13. Discuss the motion of a particle executing two simple harmonic motions in

perpendicular directions with same period.

14. A square hole is punched out of a circular lamina of radius as its diagonal. Show

that the distance of Centre of gravity of the remainder from the centre of the circle

is  a/(4π-2).

15. Derive the pedal p-r equation of a central orbit.
16. If the law of acceleration is and the particle is projected from an

apse at a distance c with velocity , prove that the equation of the orbit is

.

17. Find the moment of Inertia of a thin uniform parabolic lamina bounded by the

parabola  and y axis about the y-axis.

18. Derive the equation of motion of a rigid body about a fixed axis.

 

 

PART –C

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

 

19. (a) From a solid cylinder of height h, a cone whose base coincides with the base

of the cylinder is scooped out so that the mass centre of the remaining solid

coincides with the vertex of the cone. Find the height of the cone.

 

(b) Find the centre of gravity of the arc of the cardiod r=a(1+cosθ) lying above

the initial line.                                                                                                             (10 + 10)

 

20. (a) Derive the equation to the common catenary in the form y = C cosh x/c.

 

(b) A chain of length 2l is to be suspended from two points  A and B in the same

horizontal level so that either terminal tension is n times that at the lowest

point. Show that the span AB must be

• + 10)

21. (a) A particle executing simple harmonic motion in a straight line has velocities

8,7,4 at three points distant one foot from each other. Find the period.

 

(b) Find the resultant of two simple harmonic motions of the same period in the

same straight line.                                                                                                  (10 + 10)

 

22. (a) State and prove Perpendicular axis theorem

 

(b) Show that the moment of inertia of the part of the parabola  cut off

by the double ordinate  is about the tangent at the vertex and

about its axis, M being the mass.                                                               (6 + 14)

 

 

 

 

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