Grand Valley State University The Padnos School of Engineering EXPERIMENT THREE HYDROSTATIC FORCES EGR 365 Fluid Mechanics Julie Watjer May 20 2003 Julie Watjer Page 2 5 20 2003 Outline 1 Purpose To experimentally determine the effect of hydrostatic force on a submerged planar surface and to compare those experimental results to theoretical predictions 2 Background Theory a Apparatus Figure 1 shows the apparatus used for this lab Figure 1 Apparatus The 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 equation i a g p c Pressure as a function of z i p z patm gz Julie Watjer Page 3 5 20 2003 d Closing moment i M R T sin 2 0 5 L mg sin 1 e Opening moment L d 2 i M gw0 5d 2 cos 1 3 cos 1 f Tension in terms of depth L d 0 5L mg sin 1 gw0 5d 2 2 cos 1 3 cos 1 i T d RT sin 2 3 Method a The tensile stress on the door was increased by adding weight b Measured the height that the gate opened corresponding to the weight added 4 Results a Calculated hydrostatic forces on apparatus b Sample calculation appendix a 5 Interpretation and Conclusion a Comparison of results theoretical to experimental 6 Error Analysis 7 Appendix a Calculations b Design Questions 1 Purpose To 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 2003 2 Background Theory 2 b Euler s Equation Pressure 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 a g p 1 2 c Pressure as a function of z If 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 p z patm gz 2 2 d Closing moment Derivation of the closing moment n M r dF i i i 1 R FT sin 90 0 5L Fw sin 90 M R T sin 2 0 5L mg sin 1 3 2 e Opening moment Derivation of the opening moment M r dF L s pwds r L s p g depth depth d s cos 1 d cos 1 M L s g d s cos 1 wds 0 M gwd 2 2 L d cos 3 cos 2 1 1 4 Julie Watjer Page 5 5 20 2003 2 f Tension in terms of depth The 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 L d 0 5 L mg sin 1 gw0 5d 2 2 cos 1 3 cos 1 T d RT sin 2 5 3 Method For 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 Results MathCAD 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 please reference 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 measured values 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 Page 6 Total Water Depth from tank bottom cm Julie Watjer 5 20 2003 Tension force in X vs Water Depth 15 10 d 5 0 5 5 10 0 5 10 5 6 1 10 1 5 10 t d Tensile force in X milli N 6 2 10 6 Figure 2 Theoretical Tensile force vs Depth Mass g 100 200 Force N mgcos 0 803588 1 607176 Depth cm 6 35 6 85 300 400 2 410764 3 214353 7 5 8 65 Table 1 Experimental Data Obtained Figure 3 Experimental Results with 10 Error Bars A sample calculation can be found in Appendix A Julie Watjer Page 7 5 20 2003 5 Interpretation and Conclusions It 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 match the 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 may have caused the …