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Sunday, 2 September 2012

EXPERIMENTAL STRESS ANALYSIS



1. Define white light and monochromatic light?
                                    Light from the source that emits a continuous spectrum with equal energy for every wave length is called white light. Light of a single wave length is called monochromatic light.

2. Define wave front?
                                    The locus of point on different radial lines from the source exhibiting the same disturbance at a given instant of time is a surface known as a wave front

3. Define ray?
             A line normal to the wave front, indicating the direction of propagation of the waves, is called a ray.

4. Define plane polarized?
                           If all the waves have their light vectors in the same direction, the light is said to be plane polarized.

5. Define refractive index?
                        The ratio of the velocity of light in air to the velocity in the medium is called the refractive index of the medium and is denoted by ‘n’

6. Define wave length?
                        The wave length of any given frequency is the distance traveled during one complete vibration. It is denoted by angstroms.

7. Define amplitude?
                        This is a magnitude of the disturbance. The intensity of the light may be taken as directly proportional to the square of the amplitude of the vibration

8. Define birefringence?
                        When a ray is incident on certain crystals like calcite or Iceland spar it is split at entry in to two components which in general are transmitted through the crystal in different directions with different velocities is known as birefringence

9. Define optic axis of the crystal?
                                    Any line which is equally inclined to the concurring edges at the two obtuse corners of the crystal is defined as the optical axis of the crystal


10. Define uniaxial crystal?
                        A plane normal to the optic axis is called as equatorial plane. Crystal having only one optic axis are called uniaxial crystals.

11. Define biaxials crystal ?
                        Many crystal having two optic axis and as such are known as bi axial crystals

12. What are the steps involved in the production of the plane-polarized light ?
                                 The following methods may be used to produce plane polarized lights are
                                           - Blackened glass plate
                                                - Pile of plates
                                                  - Calcite crystals
                                                  - The Nicol prism
                                                  - The glan Thompson polarizer
                                                  - Ahrens polarizer

13. Define is Ahrens polarizer?
                                    This is essentially a double Glan­­-Thompson prism in which, however, the ordinary ray is not supported but emerge prism from the prism and is eliminated by being diverted out of the regular cone of polarized rays. It has no lateral displacement.

14.Define temporary double refraction ?
                                    Many transparent non-crystalline materials that are optically isotropic when free of stress become optically anisotropic and display characteristics similar to crystals when they are stressed. These characteristics persist while loads on the material maintained but disappear when the loads are removed. This behavior is known as temporary or artificial double refraction.

15.Define stress optic law ?
                                    Stress optic law states that the changes in the indices of the refraction were linearly proportional to the loads.

16.state Mesnager’s theorm ?
                                    When an isoclinic cuts the stress trajectory at right angles, the principal stresses at the points are a maximum or minimum

17.what are the applications of the fractional fringe order determination ?
                                    (1) Compensation techniques
                                    (2) Colour matching techniques
                                    (3) Equidensometry method

18. State the method used in compensation techniques ?
                        1.The Babinet compensation method
                        2.The babinet soleil compensation method
                        3.Tension or compression strip method
                        4. Tardy method of compensation
                         5. Senarmont method of compensation
                        6. Photometric method


19. What are the steps involved in the senarmont method of compensation ?

            The following steps are involved in this method
                1. Remove first quarter plate.
                2. Rotate the system of analyzer and the polarizer
              3. Rotate second quarter wave plate until one axis is parallel to the axis of the polarizer
               4. Rotate the analyzer until extinction is obtained at the point of interest.

20. Define photometric method ?
                              A serious disadvantage of the tardy and the Friedel’s method of compensation is that they involve a sequence of operations and are, therefore, suitable only when the state of stress is static. By means of the photometric method is also known as the intensity method.

21. Define Saba tier effect ?
                                    Pseudo-solarization or Saba tier is the partial reversal of the image obtained when the film is given a limited exposure to uniform light during development and when thus consequently the less exposed areas of the negative are preferentially darkened to a certained extent.

22. Define Mackie line?
                        The lines which are in contrast with results expected from the characteristics of the material is known as Mackie lines.



23.Define calibration techniques ?
                                    The photo elastic material has to be calibrated to determine the material fringe value so as to convert the fringe orders in to stresses. The following methods may be used to calibrate a photo elastic method…..
                        1. Simple tensile specimen
                        2. Beam under pure bending
                        3. Circular disc under diametral compression

24. What are the methods of the separation of the principal stresses ?
                        1. Methods based on the equilibrium equations
                        2. Methos based on the compatibility method
                        3. Methods based on the hooks law.

25. Define time-edge effects?
                        When a photo elastic material is machined and examined under a no load condition as a function of time it is noted that a stress is induced on the boundary which produces a series of fringes which are parallel to the boundary of the model. These fringes super impose on the fringes obtained by applying load to the model and thus produce errors in the photo elastic data. This phenomenon is known as time-edge effects.

26. Define epoxy resins ?
                        These are the considerations products of the epichlorohydin and polyhydric phenol. They are suitable for two-dimensional photo elastic work. The finished resin is formed by the chemical reaction of the basic resin  and the hardener.

27.Define Columbia resin CR-39 ?
                                    This is allyl diglycol carbonate which is produced by reacting phosgene with diethylene glycol to obtain a chloroform ate, which is the esterifies with ally alcohol to yield a monomer. It is brittle and cannot be machined but can be easily routed., It has relatively low fringe value and sensitivity index.

28. What are the methods of slicing?
                        1. Plate method
                        2.Difference method.

29.Define the plate method of the slicing ?
                                    In this meted the thin plate are actually sawed out of the model and each plate is examined in the Polaris cope.


                                                                                                                                                           
30. Define the difference method of slicing ?
                                    This method consisting in removing the thick slice first and then gradually reducing the thickness and difference between the fringe orders at the point between the successive patterns is taken to represents the fringe orders or stress in a plate of thickness equal to that of the material removed.

31. State two precautions for applying coating ?
                                    1. for a long term test or in particularly hostile environments, carefully clean the surface before applying any protective coating.
                                    2. When several layers of coating are required extend each overcoat beyond the previous layer.

32. What are steps followed while applying the strain gauge to the test specimen?
                        The following are the steps in general may be followed while applying the strain gauge to the test specimen are, surface preparation, gauge preparation, adhesive preparation, gauge installation, lead wire connection, environmental protection.

33.Define temperature compensation ?
                        The ideal strain gauge would change resistance in accordance with the stress-producing deformation in the structural to which it was bonded and for no other reason. Gauge resistance is affected by the many factors out of which temperature compensation is very important one….

34.Define total indicated strain?
                        The total indicated strain occurring  in a structure is made up of mechanical strain and the apparent strain. The mechanical strain is that produced by external forces. The apparent strain is the portion of the total indicated strain induced by thermal effects including expansion of base metal, expansion of gauge metal and the change in electrical resistance of the gauge.

35. What are the methods used in temperature compensation ?
                        Compensating dummy gauges, self temperature compensation, compensation by dissimilar gauges, compensation by similar gauges and compensation by computation.

36.Define transverse sensitivity ?
                                    The gauge in addition to measuring the strain along its axis also measures the strain transverse to it . This affect is reflected as an error in the strain gauge reading. This is known as transverse or cross-sensitivity of the gauge.

37.What are the method involved in the correction of the tranverse sensitivity  ?
                        (a)uni axial strain field.
                        (b)Biaxial strain field.
                        (c)Hydro static strain field

38.Define poisson’s ratio ?
                        Poisson’s ratio is defined as the ratio of the lateral strain to the  longitudinal strain


39.Define true gauge factor ?
                        True gauge factor is the ratio of the (strain indicated/2*strain calculated)*indicated gauge factor.

40. What are the advantage of the dual bridge?
                        The advantage of the dual bridge is that the left-hand bridge is left completely free for the strain gauges and all the adjustments for  both initial and null balance are performed on the right-hand bridge.

41. What are the types of strain gauge switching?
                                    1. Intra bridge switching.
                                    2. Inter bridge switching

42. What are the effect of lead wires?
                                    1. The presence of lead wires have the following effects on the strain measurements.
                                    i) The sensitivity of the circuit is affected.
                                    ii)Lead wire affects the calibration procedures and increases the balancing problem.
                                    iii) These may introduce another temperature-compensation problem.

43.Define the effect of the slips rings ?
                                    Slip rings are employed to provide lead wire connections when strain gauges are used on rotating machinery. Dirt collecting on the slip rings and brush jumps tend to generation of electrical noise.

44.Define calibration ?
                        Calibration of the potentiometer circuit may be achieved either by placing a shunt resistance across the strain gauge or the ballast resistances R2.


45.Define constant-current circuit ?
                        A constant-current power supply is a high impedance device which changes output voltage with changing resistive load to maintain a constant current.

46.Define Maxwell theorem for the alternating current bridges ?
                                    According to the Maxwell we assume a hypothetical cyclic current in each mesh of the network, such that the current in branch is the difference of the cyclic current in the adjacent meshes.

47.Define the sum of principal stresses ?
                        The sum of principal stresses can be determined by, if the change in thickness of the model as the result of the applied loads can be measures at the point of interest.

48.Define lateral extensometers ?
                                    The instruments comprises three unit, a horse shoe shaped gauge, a galvanometer, and a compensator element house in a metal cabinet. The cabinet comprises of batteries and switches.

49.What are the types of the optical interferometers ?
                                    1)mesnager’s interferometer
                                    2) Post’s series interferometer.

50. Define oblique incidence methods?
                                    When a light is passed through the not at a normal incidence an oblique incidence fringe pattern is obtained. For this oblique method secondary principal stresses and the stress optic law are obeyed.

51. State the process involved in the separation of the strain.
                                    By employing two isochromatic fringes pattern one from normal incidence and other from the oblique incidence it is possible to separate the stresses

52.State few steps been used for the selection of the photo elastic material ?
                                    1.Transparent to light used in the Polaris cope.
                                    2.Easily machinable by conventional means.                                                  3.It should have high optical sensitivity.
                                    4.It should have linear characteristics with respect to stress and strains.

53.Define the figure of merit ?
                        A material should have more distortion if the value of 1/fe for the material is high or low. Since the poisons ratio for most of the material varies from the 0.36 to the 0.42 the ratio q=(E/fo) is used to evaluate the merit of the material. This factor is known as the figure of merit.

54.State some photo elastic materials ?
                        Epoxy resins, Columbia resins CR-39,Homolite 100,Bakelite,Glass,Geletine, celluloid.

55. Define celluloid ?
                        It is a colloidal mixture of the nitrocellulose in camphor. It is susceptible to creep and is used for large and complicated models.

56.state the photo elastic property of the glass ?
                        It is the first material to be used for the photo elastic investigations. It is transparent, free from creep, insensitivities to the mild temperature changes, isotropic, homogeneous, inexpensive, difficult to machine and low optical sensitivity.

57.Define the photo elastics method ?
                        The photo elastic method depend up on the property of certain transparent solid by which they become doubly refractive under the action of the stresses, the magnitude of the optical effect of the bearing the definite relation to that  of the stress.

58.Who discovered the photo elastic method and when ?
                        The photo elastic method was first discovered by Sir David Brewster in 1816 in sheets 0f stressed glass.

59. Define the determination of fringe order ?
                        In order to determine the fringe order at a point in a slice and to determine the stress distribution the slice may be immersed in the immersion fluid placed in the tank and viewed in the standard circular Polaris cope for the isochromaticfringe pattern and plane Polaris cope for the isoclinic fringe pattern.

60. Define the material fringe value ?
                        The material fringe value is determined from the stress frozen pattern in the calibration model. The half fringe model is determined by using the bright setup. For fractional fringe method the tardy method of compensation is used.

61.Define the analysis technique ?
                        The analysis of the stresses of the three dimensional stress frozen model may made depending upon the geometry f the model and the information being sought in the analysis.

62. State the techniques in the analysis of the fringes?
                        1. The stresses on the free surfaces
                        2. the core method
                        3. Principal stresses in general slice

63. What are the procedures are adopting the analysis of the surfaces?
                        1. Assume that the surface stresses are desired along the lines-s.
                        2. Subslice and measure the isochromatis fringes along the perpendicular directional
                        3. Subslice to obtain a thin surface and observe normal to the surface
                        4. Calculate the principal stresses.

64.Define the core method ?
                        When surface stresses are required at a specific point are along a successions of points, core method is used. This method consisting of a cylindrical specimen containing the required surface point cut at the three-dimensional model.

65. What is the procedure in core method?
                        1.Rotate the position of the cylinder to the maximums refringence.
                        2.Measure the fringe order at the free surface for this orientation
                        3.Rotate the specimen through 90 degree
                        4. Measure the fringe order for N1 of this orientation.

66.Define the tilting stage method ?
                                    The tilting method enables to determine the directions of the principal axis at any point of the frozen slice directly, after which the slice is suitable for the measurements of the principal- stress differences….

67. Define the effective stresses.?
The effective stresses can be determined from the cube from the model at any random orientation. The cube is observed in normal incidence on its three mutually orthogonal faces and the fringe ordersNx.Ny.Nz, and the isoclinic parameters Ox,Oy,Oz are recorded.

68.Define the oblique incidence method ?
                        A hydrostatic method or tension has no effect on the relative retardation and therefore there must be any photo elastic pattern ,the retardation and the isoclinic values  are  sufficient to calculate.

69.State the method used for the separation of the principal stresses ?
                                                1.Shear difference method
                                                2.Extension of Lame-Maxwell equation of the three dimensions
                                                3.Solution by compatibility equation
                                         4.Experimental methods.

70.Define traction ?
                        The starting can be obtained photo elastically by choosing point O on the free surface or on a surface is subjected is known as traction.

71. Dfeine the multi layer technique ?
                                    The  multi layer reflection technique utilizes e model fabricated from the series of cemented thin laminations of partially aluminated photo elastic plastic. The locations of the mirror coating are arrangement so that he observations can be made in any area. It is ideally suited for the model where the stress field is symmetric.

72. Define photo elastic coating?
                        The method of photo elastic coating is also called as the method of birefrigent coatings or photo stress extends the well known photo elastic method to the measurement of the surface strains in opaque body like steel,rock..Etc..

73. Define structural inspection ?
                                    The function of the structural inspection is to determine the structural integrity of the each part of the material. Defective parts are removed during the overhaul process, it save time and money.

74. Define vapour blasting?
                        Vapour blasting is employed for special cleaning of the jobs and is accomplished by means of the specially designed equipment and material. The use of the vapour blasting is limited to parts and accomplished to the material erosion.

75. Define flux leakage ?
                        A flux leakage occurs where the lines of flux leave the surface of the material resulting in the concentration of the magnetic strength at discontinuity.

76. Define strain gauge?
                        A strain gauge may be defined as any instrument or device that is employed to measure the linear deformation over a given gauge length, occurring in the material of structure during the loading o f the structure
77. List the various type of strain gauges?
         1. Mechanical
        2. Optical
        3. Interferometric type
       4. Electrical
       5. Magnetic    
       6. Acoustical
       7. Pneumatic
       8. Scratch type
       9. Photo stress gauge
78. What are the basic characteristics of strain gauge?
                        1. Gauge length
                       2. Sensitivity
                      3. Range
                     4. Accuracy of repeatability


79. Define strain sensitivity of a gauge?
                    It is the smallest value of strain which can be read on the scale associated with the strain gauge.
                1) Deformation sensitivity
                2) Strain sensitivity
80 What are the various factors to be considered before selecting a gauge?
                        The following factors must be evaluating the strain gauge
                                                1. Readability
                                                2. Ease of mounting
                                                3. Required operator skill
                                                4. Weight
                                                5. Frequency response
                                                6. Cost
81. Limitation for simple mechanical lever magnification?
                        For strain measurements on a inner surfaces of small structures, the clearance may be insufficient for the gauge and all its auxiliary equipment
82.Limitationsof compound lever magnification ?
                        Because of their design and construction, these types of gauges are very sensitive to any vibrations that may occur in the structure under test. The natural frequency of these gauges is rather low, hence almost any structural vibrations renders the readings questionable. Size of these gauges also limits their application.


83. Define the principle and the operation of the pneumatic strain gauge ?
                        The principle of operation of an air or pneumatic gauge depends upon the relative discharge of air between a fixed orifice and a variable orifice
84. What is electrical strain gauges ?
An electrical strain gauge is a device in which a change in length produces a change in some electrical characteristics of the gauge
85. What are the classifications of the electrical strain gauges ?
                        (a) The inductance or magnetic strain gauges
                        (b) The capacitance strain gauges
                        (c) The electrical resistance strain gauges
86.Define wiedemann’s effect.
            An electromotive force is induced in the coil of the electromagnet which depends on he torsional moment acting on the  core of the electromagnet which is twisted.
87. State the factors affecting magnitude of the emf.
            1. The degree of saturation of magnetic field.
            2. The geometry of the attachment to the structure.
            3. The frequency of power supply.
            4. The size of the tube.
            5. The material of the pipes and its condition.
88.Write a short note on linear variable differential transformer(LVDT).
            It is one the most useful devices for converting a mechanical displacement into an electrical signal. It is similar to the variable-inductance transducer.
89.Write the characteristics of ideal strain gauge .
            1. The gauge should be of extremely small size so as to adequately estimate strain at a point.
2. The gauge should be of significant mass to permit the recording of dynamic strain.
90. Give the types of electrical strain gauge.
            1. Unbonded gauges.
2. Bonded gauges.
3. Weldable gauges.
4. Piezoresistive gauges.
91. Define unbounded non-metallic gauges.
            The unbounded non-metallic gauge is a mechanically actuated gauge that contains a resistance element so arranged that when one part of the gauge is displaced with respect to another there is developed a change in pressure on the measuring element of the gauge.
92. Give some uses of UNM gauges.
            Gauges of this have been used to determine displacements, loads and strains in flexible cables, airplanes, bridges, vibrating members, dynamometers and pressure gauges.
93. Define Unbonded-metallic gauges.
            The principle of the unbounded-metallic gauges is based on the change in electrical resistance of a metallic wire due to the change in tension of the wire.
94. Define bonded non-metallic gauges.
            A strain gauge using direct bonding of a non-metallic resistor element to a material in which the strain is to be measured.
95. Give some advantages foil type gauges over wire type gauges.
            1. The gauge factor is higher by 4 to 10 percent than other gauges.
            2. These gauges are easier to manufacture.
            3. These gauges can be used to measure strain of curved surfaces.
96. What are piezo-resistive strain gauges?
            Crystals of silicon, germanium, quartz and Rochelle salt show a change in resistance when deformed by applying pressure. This effect can be utilized to measure strain. Such gauges are called piezo-resistance strain gauges.
97. What are the factors should be considered while selecting the strain gauge?
            1. Gauge material.
            2. Carrier material.
            3. Grid geometry.
            4. Grid configuration.
            5. Gauge factor.
            6. Gauge resistance.
98. Define gauge factor.
The change in resistance per unit strain is defined as the gauge factor.
99. What are the qualities good gauge materials have?
            1. High gauge factor.
            2. High resistance.
            3. Low temperature sensitivity.
            4. High electrical stability.
100. what are the characteristics a good carrier materials should have?
            1. Minimum thickness.
            2. High mechanical strength. 
            3. High dielectric strength.
            4. Minimum temperature restrictions.
101. What are the desirable characteristics of the bonding cements?
            1. High mechanical strength.
            2. High creep resistance.
            3. High dielectric strength.

  1.  (i) Explain Errors and Uncertainty analysis in measurements                       
   (ii) Explain the principle and working of Marten’s extensometer                  
      2.  Explain various types of Electrical strain gauges with neat sketches           

  1. . (i) What are the qualities required for strain gauge materials?                     
(ii) Explain the mounting procedure and various types of adhesives used in bonding strain gauges                                                                               
4. .(i) Obtain the equation for strain measured by a strain gauge situated at an angular position q measured anti-clockwise from x-direction in a plane strain field                                                                                                   
(ii) Explain briefly the strain gauge circuits used for static and dynamic strain measurements                                                                                                

5. Explain the effect of a stressed model in a circular polariscope                   
      6.. (i) State and derive stress optic law                                                                 
         (ii) Explain the properties of photo elastic materials at room temperature      

7.. Explain compensation technique used by Friedel’s and Tardy methods       
b. Explain the method for separation of principal stress based on equilibrium equations                                                                                                           

8.. Explain Brittle coating method in stress analysis                                          
            9. . Explain Moirés technique     
6        a) i) Explain sensitivity and accuracy in measurements
    ii) Explain the principle and working of Huggen Begger tensometer
      b) Give the working principle used in optical extensometers
          Explain Tuckerman’s optical gauge with neat sketches

7        a) i) What is the foil type strain gauge? Explain with neat sketch
8            ii) Explain various types of strain gauge materials
b) What are the methods available for computing the strain rosette datas? Explain Analytical method

9        a) Obtain the equation of sensitivity for the wheatstone bridge circuit having ‘n’ number of gauges of equal resistance and analyse the result for various numbers of acting gauges
   b) i) In a wheatstone bridge circuit, four acting gauges are mounted at the rate of the cantilever beam. Find the increase in the output of the circuit. Cantilever beam cross section 10 x 10 mm² and 50cm carries a load of 100N. Sg=2, Ig=50milli amp, Rg=120W         
         ii) Explain Range and sensitivity of potentiometer

10    a) Explain various types of photo elastic materials and their properties at room temperature
         b) i) Explain the compensation technique used by photometric method
          ii) Explain any one of the methods used for separate the principal stresses in photo elasticity

11    a) i) Explain various methods and techniques of NDT
12        ii) Give the Applications of NDT
      b) Explain about Fiber optic sensors
                                                               





                    

Applied Physics


                                                PART – A                       (10 x 2 = 20)
Answer All the Questions
1.     Among ice, water and steam which has the highest conductivity? Why?
2.     Define coefficient of thermal conductivity.

3.     Explain the term anastigmat in optics.

4.     What is chromatic aberration?

5.     Define intensity  level of sound give its unit.

6.     A hall has volume of 1.3 x 105 m3.  It has a reverberation time of 1.4 second.  What is the average absorption coefficient of the surface if the total absorbing surface is 2500 m2?

7.     Explain the neutral surface of the beam.

8.     When the length of the cantilever is 50 cm, its depression is about 20 mm for a given load; calculate the depression for the same load when the length of the cantilever is 40cm.

9.     Give the physical significance of wave function.

10.   An electron is bound in one dimensional infinite well of width 1 x 10-10 m.  Find the energy values in the ground and first excited states.
PART – B                       (5 x 12 = 60)
Answer All the Questions
11.   Describe the forbe’s method to determine the thermal conductivity of good conductors.
(or)
12.   Explain the thermal conduction through compound media
        (i) Bodies in series
        (ii) Bodies in parallel

13.   Enumerate the chief defects of spherical lenses.  Find the condition for achromatism of two thin lenses when they are in contact.
(or)
14.   Derive the condition for the longitudinal spherical aberration to be minimum in the case of a pair of co-axial lens system separated by a distance d.

15.   Describe the rate of growth and rate of decay of sound in an auditorium and derive the Sabine’s formula for reverberation time.
(or)
16.   Define absorption coefficient.  Discuss the Various factors affecting acoustics of buildings and how these can be rectified.

17.   Derive an expression for the internal bending moment of a beam in terms of radius of curvature.
(or)
18.   Explain the depression of a cantilever and derive the expression for young’s modulus of the beam fixed horizontally at one end and loaded at the other end.

19.   Derive the schroedinger time independent and time dependent wave equations.
(or)
20.   (i) Explain with neat sketch the experimental verification of matter waves using Davission-Germer experiment.
        (ii) Calculate the De-Broglie wavelength of an electron accelerated by a potential difference of 150 V.

Engineering Mathematics - I


                                                PART – A                       (10 x 2 = 20)
Answer All the Questions

1.     State any two properties of eigen values of a matrix.
 
2.     Use Cayley-Hamilton theorem to find the inverse of the matrix

3.     Find the coefficient of xn in  the expansion of

4.     Find the coefficient of xn in  the expansion of
log (1 + x + x2 + x3 + x4)

5.     Find the radius of curvature for the curve x = at2, y = 2at.

6.     Write the formula to find the center of the circle of curvature and equation to the circle of curvature.

7.     Write the tailor’s series expansion of f(x, y) about the point (a, b).

8.     State the conditions for f(x, y) to have a maximum or a minimum value.

9.     Solve (D2 + 9) y = sin 3x, where .

10.   Write the Euler’s homogenous linear differential equation of order n.
PART – B                               (5 x 12 = 60)
Answer All the Questions

11.   (a) Find the Eigen values and Eigen vectors of the matrix

(b) Diagonalise the matrix A given above by similarity transformation.
(or)
12.   (a) Find the inverse of the matrix  by using Cay;ey-Hamilton theorem.

        (b) Obtain an orthogonal transformation, which will transform the quadratic form 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form.

13.   (a) If p-q is small compared to p or q, show that
        (b) Show that the coefficient of xr in the expansion of is . Hence, show that
(or)
14.   (a) Find the sum of

        (b) Show that

15.   (a) Prove that if the center of curvature of the ellipse at one end of the minor axis lies at the other end, then the eccentricity of the ellipse is .

        (b) Obtain the equation of the evolute of the curve: x = a(cosq + q sinq); y = a(sinq - q cosq).
(or)
16.   (a) Find the envelope of the family of straight lines
y = x tan a + 2 sec a.

(b) Prove that the evolute of the tractrix
x =a(cos t + log(tan)), y = a sint is a catenary.

17.   (a) Find and classify the extreme values, if any, of the function f(x, y) = y2 + x2y + x4.

        (b) A rectangular box open at the top is to have a capacity of 108 cubic meters. Find the dimensions of the box requiring least material for its transaction.
(or)
18.   (a) Determine the points on the ellipse, defined by the intersection of the surface x + y = 1 and x2 + 2y2 + z2 = 1 which are nearest to and farthest from the origin.

        (b) Find the maximum and minimum distance from the origin to the curve 5x2 + 6xy + 5y2 – 8 = 0.

19.   (a) Solve
        (b) Solve the simultaneous equations
(or)
20.   (a) Solve xy// - 2 (x + 1) y/+ (x + 2) y = (x – 2) e2x,by the method of reduction of order.

(b) Solve by the method of variation of parameters:
y// - 2y/ + 2y = ex tan x.

Engineering Graphics


PART – A                       (10 x 2 = 20)
Answer All the Questions
1.       What is meant by orthographic projection?
2.       State the top and front views of a vertical cylinder of diameter 50     mm and height 70 mm resting on HP with one of its bases.
3.       A rectangular prism of height 15 cm with ends of size 6 x 10 cm        is to be developed completely. What will be the shape of the     development?
4.       What are critical points in the intersection of surfaces?
5.       Mention any two advantages of isometric projection.
6.       Name any two methods of drawing the isometric projection of a     circle.
7.       When will the size of the perspective be greater than that of the     object?
8.       State the uses of perspective drawings.
9.       What is the R.C.C. mix used for lintel and ground floor concrete?
  10.     What is weathering course? Give the details of it used for house                     building.

PART – B                       (5 x 12 = 60)
Answer All the Questions
11.   Draw top, front and side view of the object given below. Indicate         major dimensions.


                                       
                                        (or)
12.   (a)    Describe the method of obtaining the orthographic      projections of an object with example.                                       (6)
        (b)   State the conventions to be followed while drawing dotted lines.                                                                                             (6)

13    A cone of base diameter 60 mm and altitude 70 mm resets on its         basis on the HP. It is cut by a plane perpendicular to both the HP         and the VP and passing through the cone 10 mm to the left of the         axis of the cone. Draw the development of the lateral surface of         the right portion of the cut cone.
                                        (or)
14.   A cylinder of diameter 40 mm pierces through another vertical         cylinder of diameter 40 mm. The axes are at right angles and are         separated by 6 mm. The axis of the vertical cylinder is nearer to         the VP. The axis of the penetrating cylinder is parallel to both the         HP and the VP. Draw the projections of the cylinders and show the curves of intersection.
                                                       
15.   A hemi – spherical vessel of diameter 90 mm is placed centrally         over a cylinder of diameter 60 mm and height 75 mm which, in turn, is kept centrally over a square prism of base side 80 mm and height 20 mm. Draw the isometric projection of the disposition of      the solids.
                                        (or)
16.   A cone of base diameter 50 mm and height 55 mm is resting on its base on the HP. It is cut by a plane perpendicular to the VP         and inclined at 30° to the HP. The plane meets the axis at a         distance of 25mm from the apex. Draw the isometric view of the    truncated cone.

17.   A hexagonal prism of base 20 mm and axis length 50 mm rests on the ground plane on one of its rectangular faces with its axis     inclined at 30° to the picture plane. A corner of the base is       touching the picture plane. The station point is 60 mm in front of   the picture plane and lies in a central plane that bisects the axis.      The station point is 40 mm above the ground plane. Draw the      perspective view of the prism.

                                        (or)
18.   A frustum of a hexagonal pyramid of base edge 24 mm, top edge       12 mm and height 45 mm is placed on the ground with a base edge parallel to the picture plane and 10 mm behind the picture   plane. The observer’s eye is 50 mm in front of the picture plane    and 75 mm above the ground. The eye lies in a central plane that        passes through the center of gravity of the frustum. Draw the   perspective view of the frustum.

19.   The line diagram of a village library building is shown below.    Draw the top view of the building taking section at window – sill         level. Also draw the front view of the building. Partition walls are        of 100 thick and main walls are of 200 thick. Assume suitable       data wherever necessary.
                                  
                                        (or)
20.   The line diagram of a residential building is shown below. Draw        to a suitable scale, front view, top view and sectional view taken    along AA. Make suitable assumptions. The following standard     dimensions for doors and windows may be taken:
        D – Panelled door – 900 x 1950
        D1 – Door – 800 x 1950
        D2 – Door – 800 x 1800
        W – Window – 1200 x 1000
        V – Ventilator – 800 x 600