ME495 Lab 07 Hydrostatic Force

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Hydrostatic Force ME 495: Mechanical and Thermal Systems Lab Section 05: Thursday Group D Authors: Soeung Khanitha, Smith Emilee, Sichantha Jack, Taylor Charles Instructor: Dr. Hamid Nourollahi Thursday, February 22, 2024

1 Table of Contents Objective of Experiment ( Khanitha Soeung ) .......................................................................................... 2 Equations and Symbols (Emilee Smith) ..................................................................................................... 2 Equipment ( Khanitha Soeung ) ................................................................................................................. 3 Experimental Setup ( Khanitha Soeung, Jack Sichantha) ....................................................................... 4 Experimental Procedure ( Khanitha Soeung ) ........................................................................................... 4 Experimental Results (Emilee Smith) ........................................................................................................ 5 Discussion of Results (Charles Taylor) ....................................................................................................... 7 Lab Guide Questions .................................................................................................................................... 9 Conclusion (Jack Sichantha) ....................................................................................................................... 9 References ( Khanitha Soeung ) .................................................................................................................. 9 Appendix ( Jack Sichantha) ...................................................................................................................... 10 List of Tables Table 1: Symbols ............................................................................................................................................ 2 Table 2: Equations .......................................................................................................................................... 3 List of Figures Figure 1: Diagram of Hydrostatic Pressure Apparatus .................................................................................. 4 Figure 2: Depth vs Mass Plot ......................................................................................................................... 5 Figure 3: Plot of Depth of Pressure (Experimental and Theoretical) vs Mass ............................................... 5 Figure 4: Plot of Moment (Experimental and Theoretical) vs Mass .............................................................. 6

2 Objective of Experiment ( Khanitha Soeung ) This experiment is designed with two primary objectives. The first objective is to investigate the hydrostatic thrust exerted on a plane surface submerged in water under the varying degrees of submersion, both partial and full. The second objective is to determine the precise position of the line of action of this thrust and compare it against the theoretical expectations. In order to facilitate this investigation, the F1-10 Hydraulics Bench and F1-12 Hydrostatic Pressure Apparatus, developed by Armfield Limited in Ringwald, Hampshire, England are utilized. The team hypothesizes that the hydrostatic force will increase in direct proportion to the depth of the submersion. Furthermore, the team anticipates that the experimental determination of the line of action of the force will closely correspond to the theoretical predictions. Therefore confirming the fundamental principles governing hydrostatic force. Equations and Symbols (Emilee Smith) Table 1: Symbols Symbol Definition/Units Units D Height of end face (m) Meter (m) B Width of end face (m) Meter (m) L Length of arm (m) Meter (m) H Height of pivot (m) Meter (m) d Depth of water from base of quadrant (m) Meter (m) F Hydrostatic Thrust (N) Newton (N) m Mass (kg) Kilogram (kg) g Acceleration due to gravity Meters per Second Squared (m/s^2) w Weight Newton (N) A Area (m^2) Square Meter (m^2)

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3 M Moment (Nm) Meter (m) h Mean depth of immersion (m) Meter (m) h’ Depth of the center of pressure (m) Meter (m) h” Distance of center of pressure below the pivot Meter (m) Table 2: Equations Equation 1: Hydrostatic Thrust (partially submerged) 𝐹 = ρ𝑔?ℎ = (1/2)ρ𝑔?? 2 Equation 2: Experimental Depth of Pressure (partially submerged) ℎ" = (2𝑚?)/(ρ?? 2 ) Equation 3: Theoretical Depth of Pressure (partially submerged) ℎ" = 𝐻 − (?/3) Equation 4: Hydrostatic Thrust (fully submerged) 𝐹 = ρ𝑔?𝐷(? − (𝐷/2)) Equation 5: Experimental Depth of Pressure (fully submerged) ℎ" = (𝑚?)/(ρ?𝐷(? − (𝐷/2))) Equation 6: Theoretical Depth of Pressure (fully submerged) ℎ" = 𝐷 2 12 +(?−?/2) 2 ?−𝐷/2 + 𝐻 − ? Equation 7: Moment ? = 𝐹ℎ” Equipment ( Khanitha Soeung ) ● Armfield Limited F1-10 Hydraulics Bench ○ Facilitates experiments in fluid mechanics and hydraulics ● Armfield Limited F1-12 Hydrostatic Pressure Apparatus ○ Facilitates the measurement and analysis of hydrostatic forces on submerged surfaces ● Various weights ○ Used to increase and decrease the hanging mass of the balance arm. Multiple masses from 10 grams to 100 grams.

4 Experimental Setup ( Khanitha Soeung, Jack Sichantha) Figure 1: Diagram of Hydrostatic Pressure Apparatus The dimensions of B & D of the quadrant end-face, as well as the distances H & L, were measured and recorded for future reference. The empty F1-12 flotation tank was positioned on the F1-10 Hydraulics Bench or another suitable level surface, and the screwed feet were adjusted until the built in circular spirit level indicated that the tank was leveled in both planes. The balance arm was then positioned on the knife edges, ensuring the free swinging motion. In addition to this, the empty weight hanger was located in the groove at the end of the balance arm. The counterbalance weight was then moved until the balance arm was horizontal, as indicated by the central index mark on the beam level indicator. Experimental Procedure ( Khanitha Soeung ) The bench was checked for proper leveling using a spirit level. After which a small mass, 50g, was added to the weight hanger. Tap water was then obtained and used to fill up the flotation tank. The tank was then filled with water until the balance arm rose. Taking care that the balance arm was not to be wet or the quadrant above the water level. Water addition continued until the balance arm became horizontal by aligning the flat of the balance arm with the central mark on the level indicator. Once horizontal, the depth of immersion was read from the scale on the face of the quadrant. This process was repeated multiple times for multiple load increments by adding more weight to the weight hanger. This

5 procedure was continued until the water level reached the top of the upper scale on the quadrant face. Finally the process was redone and reversed by deducting the weights from the weight hanger. Experimental Results (Emilee Smith) Figure 2: Depth vs Mass Plot Figure 3: Plot of Depth of Pressure (Experimental and Theoretical) vs Mass

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6 Figure 4: Plot of Moment (Experimental and Theoretical) vs Mass Sample Calculations: Constants Height of End Face: 𝐷 = 0. 1 𝑚 Width of End Face: ? = 0. 075 𝑚 Length of Arm: ? = 0. 275 𝑚 Height of Pivot: 𝐻 = 0. 195 𝑚 Partially Submerged Equation 1: Hydrostatic Thrust 𝐹 = (1/2)ρ𝑔?? 2 =) = (1/2) * 997 * 9. 81 * 0. 075 * 0. 052 2 = 0. 992 ? Equation 2: Experimental Depth of Pressure ℎ" ?𝑥? = (2𝑚?)/(ρ?? 2 ) = (2 * 0. 05 * 0. 275)/(997 * 0. 075 * 0. 052 2 ) = 0. 136 𝑚 Equation 3: Theoretical Depth of Pressure ℎ" ?ℎ?? = 𝐻 − (?/3) = 0. 195 − (0. 052/3) = 0. 178 𝑚

7 Fully Submerged Equation 4: Hydrostatic Thrust 𝐹 = ρ𝑔?𝐷(? − (𝐷/2)) = 997 * 9. 81 * 0. 075 * 0. 1 * (0. 118 − (0. 1/2)) = 4. 988 ? Equation 5: Experimental Depth of Pressure ℎ" ??? = (𝑚?)/(ρ?𝐷(? − (𝐷/2))) = (0. 25 * 0. 275)/(997 * 0. 075 * 0. 1 * (0. 118 − (0. 1/2))) = 0. 135 𝑚 Equation 6: Theoretical Depth of Pressure ℎ" ?ℎ?? = 𝐷 2 12 +(?−?/2) 2 ?−𝐷/2 + 𝐻 − ? = 0.1 2 12 +(0.118−0.118/2) 2 0.118−0.1/2 + 0. 195 − 0. 118 = 0. 157 𝑚 Equation 7: Moment ? ?𝑥? = 𝐹ℎ” ??? = 0. 992 * 0. 136 = 0. 135 ?𝑚 ? ?ℎ?? = 𝐹ℎ” ?ℎ?? = 0. 992 * 0. 178 = 0. 176 ?𝑚 Discussion of Results (Charles Taylor) Figure 2: Depth vs. Mass plot shows us the relationship between mass and depth in this experiment. It shows a function that somewhat resembles a linear absolute value function. This makes sense because for all mass values left of the 0.45kg we were increasing the mass while the opposite was true while the depth was decreasing. Figure 3: Plot of Depth of Pressure (Experimental and Theoretical) vs Mass shows us the relationship between Depth of Pressure and Mass of the weights. From this data we can see the error of the depth of pressure when the experimental and theoretical values are compared. This graph shows us that the difference is generally far greater at the lower values of mass and

8 therefore, at the lower values of the depth and volume of water in the tank. Then we know that the water is applying a hydrostatic pressure to the quadrant and therefore we know that the lower this force the difference between theoretical and experimental values will increase. Figure 4: Plot of Moment (Experimental and Theoretical) vs Mass shows us the relationship between Moment and mass in this experiment. This again, gives us values for both theoretical and experimental moments. This also, in conjunction with Figure 2, shows us the way in which the volume of water in the tank and the moment applied to the quadrant are correlated. This data set depicts that the highest moment forces generally occur at the points of most mass. It is also important to note that the rate of change of the function in this figure has a discontinuous increase at 0.15 kg which is exactly where the block became fully submerged. As discussed earlier Figures 3 and 4 show us the best depiction of error because they show deviation from expected values. The error in this lab likely comes from a mix of both systematic and random errors. The error having both lack of precision and accuracy is what shows us that assumption. The possible sources of random error in this data are human measuring error with the ruler, human accuracy error with the judgment of equilibrium and measurement of the dimensions of the machine without fluid. The possible sources of systematic error are the human error of not calibrating the zero mass to equilibrium perfectly and the table not being exactly level (this is systematic error because we did not move the equipment around throughout the lab, if we did this would engender random error instead). The team hypothesized that the hydrostatic force will increase in direct proportion to the depth of the submersion. Furthermore, the team anticipates that the experimental determination of the line of action of the force will closely correspond to the theoretical predictions. The

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9 experimental data proved this hypothesis to be true. Figure 4 showed us that the force increased with mass. Lab Guide Questions Not Applicable for this Lab. Conclusion (Jack Sichantha) The main objective of this lab was to get a better understanding of hydrostatic pressures and forces acting on a submerged plane. While also determining the thrust line of action’s position, then comparing the experimental results to the theoretical. Using the F1-10 Hydraulics Bench and F1-12 Hydrostatic Pressure Apparatus, data of the water level, mass, and geometry of the apparatus, hydrostatic thrust and depth of pressure are determined. The resulting data is then graphed to show the relationship between the experimental data vs the theoretical data. Depth of water Vs mass graph shows the direct relationship between more mass requires more water to push. The Depth of pressure vs Mass graph and Moment vs Mass graph, compare the experimental data to the theoretical data, showing an error in the results. They somewhat follow the same trend, but the experimental data is slightly lower, resulting from the errors in the experiment. References ( Khanitha Soeung ) [1] Nourollahi, A. (2024). ME-495 Laboratory Exercise – Number 6 –Vibration Analysis In ME Dept, SDSU – Nourollahi. SDSU Publishing [2] Nourollahi, A. (2024). ME-495 Course Introduction_and Syllabus Spring 2024-1. In ME Dept, SDSU – Nourollahi. SDSU Publishing

10 Appendix ( Jack Sichantha)