Outline1. Purpose2. Background/Theory2-b Euler’s Equation2-c Pressure as a function of z2-d Closing moment2-e Opening moment2-f Tension in terms of depth3. Method4. Results5. Interpretation and Conclusions6. Error AnalysisGrand Valley State UniversityThe Padnos School of EngineeringEXPERIMENT THREE – HYDROSTATIC FORCESEGR 365 – Fluid MechanicsJulie Watjer May 20, 2003Julie Watjer Page 2 5/20/2003Outline1. PurposeTo experimentally determine the effect of hydrostatic force on a submerged planarsurface and to compare those experimental results to theoretical predictions.2. Background/Theorya. Apparatus, Figure 1 shows the apparatus used for this lab.Figure 1: ApparatusThe apparatus shown in Figure 1 was a fish tank that had a wall in the middle at an angle of 9° with a rectangular hole in it 10.4 wide by 14.3 cm high. The hole was covered with a door with length 16.5 cm and a mass of 149.5 g which opened out from the 9° angle. An eye hook was screwed into the door so a torque could be applied by a pulley system to keep the door shut as pressure builds until the door is force open by the hydrostatic pressure. The moment can be derived for both the opening of the door (3) and closing of the door (4) by starting with the definition of a moment.b. Euler’s equationi.pga c. Pressure as a function of zi.gzpzpatm)(Julie Watjer Page 3 5/20/2003d. Closing momenti.)sin(5.0)sin(12mgLTRM e. Opening momenti.)(cos3)cos(5.01212dLdgwMf. Tension in terms of depthi.)sin()sin(5.0)(cos3)cos(5.0)(211212RTmgLdLdgwdT3. Methoda. The tensile stress on the door was increased by adding weightb. Measured the height that the gate opened corresponding to the weight added4. Resultsa. Calculated hydrostatic forces on apparatusb. Sample calculation (appendix a)5. Interpretation and Conclusiona. Comparison of results, theoretical to experimental6. Error Analysis7. Appendix a. Calculationsb. Design Questions1. PurposeTo experimentally determine the effect of hydrostatic force on a submerged planar surface and to compare those experimental results to theoretical predictions.Julie Watjer Page 4 5/20/20032. Background/Theory2-b Euler’s EquationPressure is the normal force acting perpendicular to a surface. If viscous stresses can be neglected, which they are in this case, a force balance gives Euler’s equation (1) for pressure.pga 2-c Pressure as a function of zIf there is no relative motion then acceleration equals zero and viscous stresses can be neglected. Therefore the partial can become the full differential because there is only movement in the z direction. This can then be separated and integrated to yield the following equation (2) for pressure as a function of depth, z.gzpzpatm)(2-d Closing momentDerivation of the closing moment )sin(5.0)sin()90sin(5.0)90sin(121mgLTRMFLFRdFrMwTniii2-e Opening momentDerivation of the opening moment)(cos3)cos(2)cos(()(()cos(*)()(1212)cos(0111dLgwdMwdssdgsLMsddepthdepthgpsLrpwdssLFdrMd(3)(4)(1)(2)Julie Watjer Page 5 5/20/20032-f Tension in terms of depthThe time that the gate opened due to hydrostatic pressure the opening moment and closing moment were equal to each other. (3) and (4) could then be set equal to each other and T could be solved for in terms of d, depth. (5) is (3) and (4) combined and solved for T, tension.)sin()sin(5.0)(cos3)cos(5.0)(211212RTmgLdLdgwdT3. MethodFor the experiment the tensile force on the door, (the weight on the end of the string) was incremented from 100 grams to 400 grams in 100 gram intervals; therefore 4 data points were taken. The water depth on one side of the door was increased very slowly to see at what point the door would open. As soon as the door opened one person made a mark on the side of the tank to note at the point that the door gave way. The water heights were then documented along with the corresponding weight that was used for the trial. The height that was measure was relative to the bottom of the door and not relative to the bottom of the tank. Table 1 shows the results acquired.4. ResultsMathCAD was used to find the tensile force in relation to depth. The numbers given were substituted in for the values and tension in terms of depth was then solved for pleasereference Appendix A for the calculations. The plot of this theoretical curve can then be seen in Figure 2. These theoretical values can then be compared to the graph of measuredvalues shown in Figure 3. The values obtained during the experiment can be seen in Table 2. This table was the used to create the experimental graph shown in Figure 3.(5)Julie Watjer Page 6 5/20/200351050 510511061.51062106051015Tension force in X vs Water DepthTensile force in X (milli N)Total Water Depth from tank bottom (cm)dt d( )Figure 2: Theoretical Tensile force vs. DepthMass(g)Force (N)mgcos(θ)Depth(cm)100 0.803588 6.35200 1.607176 6.85300 2.410764 7.5400 3.214353 8.65Table 1: Experimental Data ObtainedFigure 3: Experimental Results with 10% Error BarsA sample calculation can be found in Appendix AJulie Watjer Page 7 5/20/20035. Interpretation and ConclusionsIt can be seen from Figure 2 and Figure 3 that the experimental and theoretical results do not match well at all. The experimental results show that the tension force related to depth of water is greater. There is more tensile force per cm of water then in the theoretical calculations. This is not what was expected. The experimental data did matchthe theoretical in the fact that as tension is increased (more weight) more fluid was needed to open the door (higher depth). As noted by Figure 2 and Figure 3 the concavity of the curves did not align very well and there is quite a bit of discrepancy noted.Since the hinge was loose it was expected that the results would fit the data more accurately, this did not seem to be the case. It may be that the pin was to loose and caused binding when the door opened, this may be why the experimental results did not match the theoretical results as well as expected. The foam around the opening to the gate helped to prevent most of the leaking, but it