On the use of the Cornu Method for determining Young s Modulus & Poisson s Ratio.

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1 On the use of the Cornu Method for determining Young s Modulus & Poisson s Ratio. Jai Kai Chai Shane Duane ID: SF Theoretical Physics November 11th, 2009 Fergal Haran Abstract The aim of this experiment was to determine two properties of the material known as Perspex, namely its Young s modulus and Poisson s ratio. This was done using the Cornu method of optical intereferometry to observe the strains present in a loaded beam of Perspex. Values of Y = 2.3 ± 0.6 GPa and σ = ± were obtained for Young s modulus and Poisson s ratio, respectively. The value for Y is in good agreement with the range for various forms of perspex of GPa [1]. However, the value for σ is not quite within the accepted value of σ = 0.35 [2]. This may be either because the Poisson s ratio depends on the type of perspex used, or because human error crept into the experiment. Introduction & Theory Young s modulus for a material is defined as the ratio of applied stress to strain, or Y = P ε, (1) where P is the applied stress. We define ε x and ε y as the longitudinal and lateral strains, respectively. If a perspex beam of rectangular cross-section is loaded as in Fig. 2, then its internal bending moment is given by Y Ak 2 R 1, where A is the cross-sectional area of the beam and k is its radius of gyration. In Figs. 1 & 2 the curvature of the beam has been exaggerated for clarity. We know that k = b/ 12 for a bar of rectangular cross-section. Now at 1

2 equilibrium the internal and external bending moments are equal, and we have mgl = Y Ak2 R 1, where m is the mass of each of the suspended weights and g is the gravitational acceleration. This equation can be rearranged to give Y = mglr 1 Ak 2. Now, since A = ab and k 2 = b/12, this equation becomes Poisson s ratio is defined to be Y = 12 mglr 1 ab 3. (2) σ = ε y ε x. (3) Figure 1: View of Deformed Beam with Angle φ When a beam of perspex is loaded as in Fig. 2 it has a longitudinal curvature R 1 and forms an arc of angle φ, as shown in Fig. 1. From the figure we see that the unloaded beam has length c and is elongated by c. From the definition of the angle, it is clear that φ = hence the longitdinal strain is c + c R 1 + b/2 = c c R 1 b/2, ε x = c c = b 2R 1. (4) 2

3 The expression for the lateral strain is found in a similar fashion, and is ε y = a a = b 2R 2, (5) where R 2 is the radius of curvature in the lateral direction and a is the length of the beam s cross section. Combining (4) and (5) in light of (3), we find that σ = R 1 R 2. (6) In order to measure R 1 we employ Cornu s method, which involves placing a glass plate on the deformed beam and observing the interference patterns formed by light reflecting off the upper surface of the plate & rays reflecting off the upper surface of the beam. Now if d is the distance from the glass plate to the surface of the perspex in the z-direction as a function of x and y, then the equation for points of constant d is x 2 R 1 y2 R 2 = 2(d d 0 ), (7) where d 0 is the distance when (x, y) = (0, 0). This equation tells us that the observed fringes form a pair of hyperbolae. Attempts were made to derive (7) by geometrical methods, yet were not successful. Thus (7) was assumed to be true. The hyperbolae implied by (7) have common asymptotes given by x 2 R 2 = y 2 R 1, making an angle θ with the x-axis, where Clearly, when y = 0, equation (7) implies that cot 2 θ = R 1 R 2 = σ. (8) x 2 = 2R 1 (d d 0 ). Now, on a fringe we have 2d = Nλ, where λ is the wavelength of the light used and N is an integer. Hence A similar argument leads to x 2 = R 1 (Nλ 2d 0 ). (9) y 2 = R 2 (Nλ 2d 0 ). (10) 3

4 Experimental Method The apparatus was set up as in Fig. 2. A mercury lamp (λ = nm in this case) was shone on to a glass slide angled at 45 to the glass plate in the figure. The fringe pattern on the plate was then observed using a travelling microscope and Vernier scale. A hyperbolic fringe pattern was observed and the positions of the N fringes in the longitudinal (x) and transverse (y) directions were measured relative to the centre of the pattern. The angle θ made by the patterns with the x axis was then crudely measured by eye using a protractor. Figure 2: Experimental Setup Results & Analysis Our results were graphed on MS Excel, as in Figs. 3 and 4. The error bars for these graphs were too small to be displayed. Important parameters are shown below in Table 1. Hence, from (6), Also, using (2), we have Y = 12 σ = 1.17 = ± = 2.3 ± 0.6 GP a. Using the protractor, θ was estimated to be 57 ± 5.Plugging this value for θ into (8) gives a value for Poisson s ratio of σ = 0.4±0.2, which supports our value obtained from the radii of curvature. Discussion & Conclusions The value obtained for Y was in good agreement with those in the literature. However, the value for σ is about 0.3 below that quoted in [2]. This may be 4

5 due to a difference in the perspex sample used or it may indeed be down to human error. Likely causes of this error are slight habitual misreadings of the Vernier scale associated with the travelling microscpe, or inaccuracies in focussing the microscope on the fringes. Quantity Value Error R m 0.02 m R m 0.06 m a m m b m m L m m g m/s m/s 2 m kg kg Table 1: Experimental Data Figure 3: Graph of x 2 vs. N 5

6 Figure 4: Graph of y 2 vs. N References 1. Mechanical Engineering Information, 2. N. K. Mehta, Machine Tool Design and Numerical Control, McGraw- Hill Publishing, pp173 6