Investigation of the deflection of a wooden meter rule beam

Student I.D. Criteria Marks Achieved Abstract Brief summary: state purpose of investigation/brief method/brief theory/key results/major conclusions 5 Introduction Aims/objectives, hypothesis, variables 3 Appropriate theory including how any relationship will be used in data analysis 4 Procedure List of apparatus including dimensions/appropriate labelled diagram 3 Explains measurements to be made using appropriate instruments with reference to scale/% uncertainty 4 Explains procedure clearly/control of variables-fair test/repeat measurements/demonstrates knowledge of correct measuring techniques/comments on safety 5 Data Records data in table to appropriate precision and units/ readings include appreciation of uncertainty 3 Obtains appropriate number of measurements over appropriate range 2 Data analysis Processes and displays data appropriately in a table where appropriate including uncertainties 2 Produces graph with appropriate axes with correct units and appropriate scales 2 Plots points accurately draws line of best fit 3 Determines gradient using large triangle including units 3 Uses gradient to calculate other quantities correctly to appropriate number of significant figures 2 Calculates uncertainties correctly to appropriate number significant figures. 4 Compounds uncertainties correctly to appropriate number of significant figures 4 Conclusion States a valid conclusion or conclusions 2 Discusses final conclusion in relation to aim or aims of the experiment 2 Evaluation Discusses main source of uncertainties and/or systematic error 2 Discusses realistic suggestions to reduce error to improve experiment/ suggests relevant further work 2 Report Report is structured using appropriate subheadings/ few grammatical or spelling errors/clear on first reading 2 Appropriate references 1 Total 60 Investigation of the deflection of a wooden meter rule beam IFP Physics Lab Report Abstract In this laboratory experiment the main aim is to calculate the deflection of wooden meter rule beam and Young’s modulus and compare it to standard value. The theoretical relationship for given loads (masses from 100 g to 900 g) is linear, i.e., obeying Hook`s law. We studied the deflection of wooden meter rule beam under controlled loads with uniform loading step. The beam was supported with G-clamp and wooden blocks on one end, other was loaded with mass and the height from ground was measured, from the height measurements the deflection was calculated as difference between unloaded and loaded level, then Young’s modulus was calculated. For width and thickness measurements we calculated random error and absolute error, for the Young’s modulus we calculated absolute error (applying rules for indirect measurements error estimation). With all this processed data we plotted the graph of deflection versus mass and made a linear fit, with , which means that deflection is linearly proportional to applied mass (or loading, which is the same in our case, since , where m – mass, F – applied force to the beam, and g – free fall acceleration). The obtained value of Young`s modulus is very close to the compressed Japanese cedar. Introduction In this experiment the aims are to measure the height of the ruler of the wooden metre beam by building an apparatus consisting of eight 100g masses, two metre rules, clamp stand, G-clamp, two small wooden blocks, vernier calipers then calculate deflection, Young`s modulus, investigate of the beam`s elastic behavior, compare calculated Young’s modulus to the standard value and make conclusions about precision of measurement and possible improvements. We can assume that the wooden metre rule beam is the textbooks known problem of end-loaded cantilever beam. The deflection of the end-loaded cantilever beam is given by: (1) where – deflection, m; – load, N; – length of the beam, m; – Young’s modulus, Pa; – area moment of inertia of the beam's cross section, . Area moment of inertia is given by: (2) where - thickness of the beam, m; – width of the beam, m. Substitute (2) in (1) we get: (3) Since the deflection would be calculated as difference between “0” level position (unloaded beam) and ground, L, h and b are measured and force calculated as (g-free fall acceleration, m - mass), so the only unknown value is the Young’s modulus, we can express it from (3) as: (4) It is hypothesized that under given loading conditions (masses from 100 g to 900 g) the wooden beam would experience only linear elastic behavior, so the relationship (4) would hold and with increasing mass, the deflection would also increase. Procedure Variables: Independent variables: length (L), width (w), thickness (t) Dependent variable: force (F), deflection (d), Young’s modulus (E) Controlled variable: mass (m) Apparatus and Measurements: • eight 100g masses • two metre rules • clamp stand • G-clamp • two small wooden blocks • vernier calipers Procedure and Methods: Figure 1 – Apparatus 1. The average thickness of the meter rule is measured by using a vernier caliper and the reading is recorded. 2. By using another metre rule width is measured and the reading is recorded. 3. The apparatus is set up as shown in figure 1. 4. A 100 g of mass is hung and the length of the deflection is observed. 5. The reading obtained is tabulated in a table. 6. Step 4 and 5 is repeated by using a mass of 200 g, 300 g, 400 g, 500 g, 600 g, 700 g, 800 g, 900 g. 7. A graph of deflection of the end of the ruler, d against mass, M is plotted. 8. The Young’s modulus, E of wood of the meter rule is calculated. Data and observations Length of the beam: m (g) F (N) h (cm) d (cm) 86.6 100 0.98 85.4 0.2 200 1.96 84.1 1.5 300 2.94 82.8 2.8 400 3.92 81.7 3.9 500 4.91 80.4 5.2 600 5.89 79.3 6.3 700 6.87 78.1 7.5 800 7.85 76.9 8.7 900 8.83 75.6 10 Table 1 – Measured height from ground level and calculated deflections 1 2 3 4 5 Average Uncertainty w (mm) 27.32 27.27 26.66 26.86 26.77 26.976 0.386 t (mm) 6.52 6.63 6.51 6.56 6.50 6.544 0.067 Table 2 – Measured thickness and width of the beam Calculations 1. Deflection is calculated as difference between non loaded position and loaded for each mass: was measured with meter rule, so it`s uncertainty is the same as L, . Since the deflection was calculated only from height measurements then deflection`s uncertainty is the same as height`s one: 2. Average width and thickness: 3. w and t uncertainty estimations: Student coefficient for N=5, : , then random error: Absolute error, (we take instrumental error of vernier caliper and instrumental error of rule meter ): Record these values in the table 2. 4. Young’s modulus calculation: Using (4) to calculate Young`s modulus for all deflections, the results record in the table 3: d (cm) Y (Pa) - 0.2 1.33E+11 1.5 3.54E+10 2.8 2.84E+10 3.9 2.72E+10 5.2 2.56E+10 6.3 2.53E+10 7.5 2.48E+10 8.7 2.44E+10 10 2.39E+10 Average: 3.87E+10 Table 3 – Calculated Young’s modulus 5. Calculation of uncertainty in Young`s modulus: We show calculations only for first calculated Young`s modulus, since for the average estimation it would be the same: Data analysis Graph: With the raw and processed data in the part above we can plot the graph of d vs M (Fig. 3), but uncertainties are so low (), so we can`t see them on the graph properly. Figure 3 – Deflection dependance on mass (--- – linear fit, - data). It could be seen from the graph that the relationship between deflection and mass is demonstrating linear behavior with , that is exactly what we expected from expression (3). Comparison: Since we don`t know the type of the wood, the only thing we can do is to find the closest known value of Young’s modulus and define the wood according to table value. We found that the closest value have compressed Japanese cedar. Conclusion Conclusion: The aims of this experiments were to measure the height of the ruler of the wooden metre beam by building an apparatus consisting of eight 100g masses, two metre rules, clamp stand, G-clamp, two small wooden blocks, vernier calipers then calculate deflection, Young`s modulus, investigate of the beam`s elastic behavior and compare calculated Young’s modulus to the standard value. With measured height we calculated deflections, also we measured width and thickness and calculated random and absolute error. With all this data we calculated Young’s modulus and estimated it`s error: , compared it to standard values and found that the wood with closest Young’s modulus in compressed Japanese cedar. Then we plotted deflection versus applied mass and made a linear fit with . From the linear fit we made a conclusion that theoretical dependance of deflection from mass (or applied force, since holds and equals to: The hypothesis made is accepted when mass is increased the deflection of the end of the ruler is increased too and under given loading conditions (masses from 100 g to 900 g) the wooden beam experienced only linear elastic behavior. Evaluation and Improvement: Since the wooden meter rule beam is made out of processed and compressed wood, the Young’s modulus can vary in very wide range of values, not only because of wood is anisotropic material, but also could different levels of compression significantly influence it. The main sources of errors in this experiment are measurement of height, width and thickness of the sample. They can be eliminated by using more precise measurement techniques for example optical. Also, we assumed, that the given masses didn`t have uncertainties, which they obviously have, so another step for increasing precision of this experiment is to figure out what is mass uncertainty. We didn’t find where the ruler beam don`t acts as a spring, so the further improvement of this experiment would be using more load with smaller steps, i.e. using masses of 50 g on each step after 900g to precisely define where starts non-linear zone. References • Gere, James M.; Goodno, Barry J. Mechanics of Materials (Eighth ed.) • Graph of the apparatus link: https://pdfslide.net/documents/experiment-9-youngs-modulus.html • Buan Anshari, Akihisa Kitamori , Kiho Jung, Ivon Hasse, Kohei Komatsu and Zhongwei Guan Mechanical properties of compressed wood with various compression ratios, Conference: International Symposium of Indonesian Wood Research Society At: Denpasar, Bali, Indonesis Volume: 2, nov. 2010