MEASUREMENT OF THE MODULUS OF ELASTICITY OF SCALES MADEOF DIFFERENT MATERIALS USING THE CANTILEVER BEAM EXPERIMENT by

Size: px

Start display at page:

Download "MEASUREMENT OF THE MODULUS OF ELASTICITY OF SCALES MADEOF DIFFERENT MATERIALS USING THE CANTILEVER BEAM EXPERIMENT by"

Leslie Short
5 years ago
Views:

1 MEASUREMENT OF THE MODULUS OF ELASTICITY OF SCALES MADEOF DIFFERENT MATERIALS USING THE CANTILEVER BEAM EXPERIMENT by 1: Morka J.C. Department of physics, College of Education, Agbor : Esiekpe Lawrence Ese Department of Physics, Federal College of Education (Tech), Asaba : Osuhor P.O. Department of Physics, Alizomor Secondary School, Owa- Alizomor Abstract The Elasticity modulus of scales made of different materials using the cantilever beam arrangement is presented in this study. The Elasticity modulus of the various materials was studied by varying the mass suspended and the length of beam to examine the behaviour. Also investigated is how to obtain a good approximation of the formula that is appropriate with a change in the point of suspension of the masses. The result showed a quadratic function fit on the data, which gave the best curve and is represented by f = α I x + β I x, where the fit gave α I = 0.001; β I =0.0 Introduction Young Modulus and the cantilever in solid mechanics, the slope of the stress-strain curve at any point is called the tangent modulus of the initial, linear portion young s modulus, also known as the tensile modulus. It can be defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke s law holds. It is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the materials. In anisotropic materials, young modulus may have different values depending on the direction of the applied force with respect to the material s structure. It can also be called commonly, the elastic modulus of elasticity. The S.I Unit of modulus of elasticity, E, is NM -. The elasticity modulus, E, predicts how much a materials sample extends under tension or shortens under compression. Some calculations also require the use of other material properties, such as the shear modulus helps in selection of materials for particular structural applications. The Cantilever A cantilever is a beam supported only one end. The beam carries the load to the support where it is resisted by moment and shear stress. Cantilever arrangement allows for overhanging structures without external bracing. In most applications, objects are assumed to be rigid for the purpose of simplification. When supposedly rigid materials are subject to great forces, there is a permanent deformation. When subjected to a particular stress, or force per unit area, materials will respond with particular strain. In this experiment, we considered the cantilever beam in the position shown below. Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 5

2 Morka,.J.C., Esiekpe Lawrence Ese & Osuhor P.O. y Z P f The arrangement defined coordinate Z along the length of the beam, coordinate y vertically from the center line of the beam and coordinate x width wise across the beam. Thus making the arrangement a right handed system beam deflections (in the y direction) are denoted using the variable, f, which is a function of Z i.e. f = f (Z) (1) The bending moment M x (Z) is positive if the upper fibers of the beam are in compression and the button fibers are in tension. For a symmetric cross section beam made of a linear elastic material, whose displacements and slope under load are small relative to its underfromed configuration, the relationship between the displacement and bending moment is d f EI dz M x () where EI is the bending moment of the beam, E is the modulus of elasticity and I is the second area moment of the cross section (L 4 ) about the x-axis For a rectangular cross section bh I () 1 where b = width of the material h = Thickness of the material from () and (), we obtained after some manipulations, f mgz (L Z) (4) 6EI From (4), if Z L Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 6

3 Measurement of the modules of elasticity of scales made of different materials using.. 4mgl (5) bh f E Procedure Firstly, we use a wooden scale to obtain the relation of the depression from the unloaded position for various loads attached at a particular point on the scale. We also obtained the young s modulus from this we used a travelling microscope and a pin to find the depression. This we did by subtracting the height from the unloaded reading on the travelling microscope. Secondly, we hung the plastic scale material and repeated the same procedure as above to calculate the young s modulus of the plastic scale. For the wooden scale we varied the length of the scale while keeping the point of suspension to be same and a similar procedure was carried out as before for various lengths so as to obtain the relation between the length of beam and the depression. Also, varied is the point of suspension for a particular length and the relation between length and the depression was examine to see how the parameters are related to each other. Observations and Results Dimensional measurements (a) Thickness of wooden scale h w =0.065m (b) Thickness of plastic scale h p = 0.059m (c) Width of the wooden scale b w = 0.07m (d) Width of the plastic scale b p =0.045m Table 1 Measurement readings of the stress to the depression of the wooden scale for L= 0.40m S/N Mass Suspended (M) Measured Height(M) Depression F (M) For a linear fit on the above data for m against L, the data given by GHU-PLOT was a I = where a I is the slope of the graph from where we calculated E = 4.1X10-1 N/m Table : The readings as obtained for the plastic scale S/N Mass Suspended (M) Measured Height(M) Depression F (M) Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 7

4 Morka,.J.C., Esiekpe Lawrence Ese & Osuhor P.O The slope of this was approximately The young modulus for L = 0.6m was calculated to be E = 0.61 x 10-1 N/m Table : This shows the data for the various lengths of beam S/N Length(Cm) Mass Measured Depression(Cm) Suspended(Gm) Height(Cm) From the above result, it is observed that the relation between length and depression for a given mass and depression for a given mass shows that the depression is directly proportional to the cube of the length. Table 4 S/N RATIO OF L f AT 0.05(Kg) f AT 0.10(Kg) f AT 0.15(Kg) We also carried out the analysis for the relation between the point of suspension and the depression for 0.150kg mass and with a clamp length of the beam at 0.4m 5: Shows the result in a tabular form S/N DIST OF POINT OF MEASURED HEIGHT (M) DEPRESSION, f (m) LOADING(CM) 1 Unloaded Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 8

5 * Depression (cm) Measurement of the modules of elasticity of scales made of different materials using Conclusion The result of the analysis showed a quadratic function fit on the data, whose best curve is as shown below given by the function f = α I x + β I x where the fit gave α I = 0.001; β = 0.0 The modulus of elasticity E for the wooden scale is 4.1 x 10-1 N/m while E, for the plastic scale if applied in engineering purposes Distance between point of suspension and point of loading (cm) Plotted in centimeters to avoid ambiguities in scale choice References Beer, M.F.P et al (1981). Mechanics of materials. Mc Graw-Hill Book. New York, N Y Pp Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 9

6 Morka,.J.C., Esiekpe Lawrence Ese & Osuhor P.O. Hibbeler R.C (1991). Mechanics of Materials Macmillan : Longman Publictions: New York. Landau L.D and Lifshitz E.M (1986). Course of theoretical Physics Vol. 7 Theory of Elasticity, Pergamon press, Oxford Modulus of Elasticity (010) Flexure. Experiments in Mechanics. Vishay, <http// Nelkon M, and Parker P (1984). Advanced level physics. Heinemann Educational Books:London. Riley, F.W and Storges, L.D (199). Engineering Mechanics. John Wiley and sons, New York Timoshenko, S.P (198). History of strength of materials. Dover Publications: New York.. Journal of Research in Pure and Applied Sciences Volume 5 Number 1 December 015 Page 0

Experiment Five (5) Principal of Stress and Strain

Experiment Five (5) Principal of Stress and Strain Introduction Objective: To determine principal stresses and strains in a beam made of aluminum and loaded as a cantilever, and compare them with theoretical