DETERMINATION OF ELASTIC CONSTANTS CORNU S METHOD
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1 DETEMINATION OF ELASTIC CONSTANTS CONU S METHOD Ai: To deterine the elastic constants of the given aterial by Cornu s interference ethod. Apparatus: A glass or perspe plate is placed syetrically on two knife-edges to for a bea. The weight hangers are suspended fro the bea syetrically at a distance p fro the nearby knife-edge. Equal asses are place on the weight hangers. A conve lens (to for elliptical fringes or a plane glass plate (to for hyperbolic fringes is place at the iddle of the bea. The air fil fored between the bea and the lens/glass plate is illuinated by eans of the light fro a sodiu vapor lap. The light is ade to fall norally on the air fil with the help of a glass plate arranged at 45 o as shown in the figure. The interference fringes fored are observes by eans of a traveling icroscope. Principle:- when the bea is bent under the action of a load it has a definite radius of curvature in longitudinal and transverse directions which depend upon the geoetry of the bea and Young s odulus of the aterial. The radius of curvature changes with the change of the load. Fro the study of these changes, elastic constants of the aterial of the bea can be deterined. The changes in radii of curvature in two utually perpendicular directions are investigated by optical interference ethod by foring Newton s rings (elliptical or hyperbolic fringes. A cobination of glass plate/perspe plate and a conve lens of large focal length (~00c is used to get elliptical fringes and if a plane glass plate is used instead of the conve lens, hyperbolic fringes are fored. In both of these cases. it can be shown that D n,where D is the difference in squares of diaeters or fringe separation and n is order of the fringe. When the test plate is bent uniforly it suffers a longitudinal elongation and transverse copressional strain at the sae tie. Let equal asses be suspended fro the weight hangers. p is the distance between weight hangers and nearer knife edge. If Is the longitudinal radius of curvature of the bent plate, YI gp and gp ( YI
2 where Y is the Young s odulus of the aterial and I is geoetrical oent of inertia of the bea. For another ass, if is the corresponding radius of curvature, gp ( YI (- ( YI gp (3 If and ( k are the distances between the th and the th ( k pair of dark fringes respectively in the longitudinal direction with ass, then, 4 (4 ( k 4 ( k (5 where is the wavelength of light fro the sodiu vapor lap. ( k 4 (5-(4 k 4k ( k Siilarly if and ( k (6 are the corresponding distances with ass, then, (7-(6 4k ( k 4k (7 ( k ( k 4k (8
3 where ( k fro (3 and (8, YI and = gp ( k 4 k gp Y 4k I (9 If b is the breadth and d is the thickness of the plate, then, I = Y 3 gp 3 4bd k I (0 Hence Y can be calculated. If and 3 bd. Substituting in (9, are transverse radii of curvature of the bent plate with asses and, then, 4k ( y( k y y( k y = 4 k ( where = y ( k y and = y ( k y here, y, y ( k, y and y( k are the corresponding distances in transverse direction for and. Now Poison s ratio, = Transversestrain Longitudin al = strain Changeof curvaturein Y direction Changeof curvaturein X direction
4 = = Y and σ can also be calculated by plotting a graph taking the order of fringes along the X-ais and square of the distance between corresponding fringes along the Y-ais. For any load M, we can plot two straight line graphs ( one for longitudinal distance and the other for transverse (3 distance. Let and be the angles ade by the straight lines with the X-ais in longitudinal and transverse directions for a load M. Siilarly for load M, let and be the transverse corresponding angles. Then the slope, dy d tan M 4 d dm 4Cot and Siilarly, 4Cot 4Cot (4 4Cot (5 and Hence 4Cot Cot Fro (3 Y 3 M M gp (6 3 bd Cot Cot Also, 4Cot Cot Fro (3, Cot Cot Cot Cot (7 igidity odulus n and bulk odulus k are given by
5 n and Y k Y 3 Procedure : The eperiental arrangeent is done as shown in figure. Equal asses (M each are placed on the weight hangers. Bright and dark fringes are observed through the icroscope. The icroscope is oved parallel to the length of the test plate(ie. in the longitudinal direction. The cross wire is ade tangential to the 0 th dark fringe on one side ( say left in longitudinal direction. The icroscope reading is taken. The cross wire is then ade tangential to the 8 th, 6 th, 4 th etc. and nd dark fringes on the sae side and the corresponding readings are taken. Then cross wire is adjusted tangential to the nd, 4 th,.0 th fringes on the other side (right in succession and the readings are taken in each case. The diaeters of fringes are calculated by taking the difference in the readings on the left and right sides. The square of the diaeter( is calculated and the value of ( k are deterined for a constant value of k (say k=0. Hence, is calculated. Siilar observations are recorded for transverse direction and the y y are found. Hence ( k is also calculated. The eperient is repeated for different load M and the corresponding and are calculated. The distance between weight hanger and nearer knife edge is easured as P. The breadth of the bea is found out using vernier calipers and the thickness is easured using screw gauge. Hence Young s odulus (Y and Posson s ratio( are calculated. Graphs are plotted by taking order of the fringes along X-ais and squares of distance along Y-ais. Slopes of the n vs graphs are calculated as tan and ( k substituting values of Cot in the equation for Y and, these quantities are calculated. The ean value of Y,, n and k are calculated fro the values calculated using the observed readings and the values calculated fro graphs. esults. Young s odulus Y = N/
6 Poisson s ratio, = igidity odulus, n = N/ Bulk odulus, k. = N/ Telescope S Glass plate at 45 o OBSEVATIONS.Transverse readings : M = gs. P = c. Order of the fringe Microscope reading Left(c ight(c Diaeter D(c D (c D k D (c
7 Mean D D k = = Mean ( D D k = -. Transverse readings: M = gs. P = c. Order of the fringe Microscope reading Left(c ight(c Diaeter D(c D (c D k D (c Mean D D k = = Mean ( D D k = - epeat the eperient for another M value. Breadth of the plate (Using vernier calipers Least count = M.S. V.S. Total(c
8 D ( D ( Mean breadth (b= Thickness of the plate(using screw gauge Least count = P.S. H.S. Total(c Mean thickness (d = Young s odulus, Y = Longitudinal M g Transverse M g Slope = tan Slope = tan Order (n Order (n tan = cot = tan = cot =
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