Hydrostatic Forces by Dan Schwarz School of Engineering Grand Valley State University EGR 365 Fluid Mechanics Section 01 Instructor Dr S Fleischmann May 22 2006 Outline I Purpose Statement A Experimentally determine the hydrostatic forces on a submerged planar surface B Compare experimental results to theoretical predictions II Background A The pressure variations within a volume of fluid can be described by Euler s equation a g p 1 B From Euler s equation it was determined that pressure varies linearly in a static body of water open to the atmosphere p z p atm gz 2 C In this experiment an inclined hinged door will be subjected to a hydrostatic load until it opens The experimental device is shown below D Two opposing moment equations were developed to determine when the door would open Details in appendix 1 Closing moments gravity string tension M RT sin 2 0 5 L mg sin 1 3 2 Opening moment hydrostatic force of rising water M 0 5 gwd 2 L cos 1 d 3 cos 2 1 4 3 These two equations were set equal to each other to predict water depth as a function of tension depth was predicted by finding the roots of the resulting polynomial E Experimental Method 1 Measure all dimensions and angles shown in the figure with a ruler mm and protractor degrees 2 Measure a mass kg record the measurement and hang it from the pulley 3 Place a strip of tape on the side of the tank 4 Slowly fill the tank with water and mark the water level when it reaches the bottom of the door 5 Continue filling the tank with water and mark the water level as soon as the door opens 6 Measure the difference between the initial mark and the final mark and record the measurement 7 Repeat this method for several masses III Results Discussion A The measurements taken from the experimental procedure are compared with the predicted water levels in the table below Discrepancies were initially high because the water levels did not change much before the door opened Mass kg 0 0200 0 0500 0 1000 0 2000 0 5000 0 7000 Tension N 0 1962 0 4905 0 9810 1 9620 4 9050 6 8670 Predicted Water Level m 0 0215 0 0276 0 0357 0 0484 0 0765 0 0919 Measured Water Level m 0 0420 0 0460 0 0530 0 0630 0 0910 0 1040 Discrepancy 95 62 66 91 48 67 30 30 19 00 13 23 B Water depth was plotted as a function of tension in the figure below The figure seems to indicate that the predicted depths were less than the measured depths by the same amount There may have been more moments and forces that where unaccounted for in the prediction IV Conclusions A Discrepancies were most likely caused by measurement error when measuring the water depth meniscus reaction time etc B Discrepancies may also result from forces not accounted for in the equations such as friction in the hinges V Appendices A Appendix A Derive Working Equations 1 Closing Moment Equation a Determine moment caused by the weight acting on the centroid of the door 0 5 L mg sin 1 b Determine moment caused by the tension force on the door RT sin 2 c Sum the closing moments M RT sin 2 0 5 L mg sin 1 3 2 Opening Moment Equation a Setup generic moment integral M rPdA b Write r in terms of the variable s r L yr L Ix s2 s L y c dA 6ds 2 c Write P in terms of the variable s P gs cos 1 d Write dA in terms of the variable s dA wds e Substitute r P and dA into the integral s2 s M L gws cos 1 ds 6ds 2 f Integrate with respect to s 1 s3 s3 gw cos 1 M s 2 L 6 6 2 g Substitute s d cos 1 and rearrange the results M 0 5 gwd 2 L cos 1 d 3 cos 2 1 4 B Appendix B Assigned Design Calculation 1 The tension needed to hold the door shut with the water level at the hinges was determined to be 14 3726 N 1 4651 kg using the equation T 2552 1076d 3 1081 3674d 2 0 2774 C Appendix C Assigned Design Questions 1 Loosening the hinge pins on the door would reduce the friction moment created on the door Since the hinge friction was not accounted for in the working equation reducing the friction would decrease the discrepancy 2 The foam would eliminate premature leaking at the bottom of the door which would make it easier to determine the opening depth It would also increase the opening depth because the foam produces a frictional force 3 4 points are required to fit a third degree polynomial Since we are modeling a cubic relationship we should collect at least 4 data points 4 Recording the water depth electronically would be advantageous since the depth measurement is difficult to take