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Hydrostatic Forces By Jon den Hollander EGR 365 Section 02 Dr. M. Sozen 5/27/2008 Abstract The purpose of this laboratory exercise was to determine the force on a submerged planar surface caused by a hydrostatic fluid. Water was added to a fish tank having a hanging door being held closed by a string with a tension caused by a hanging weight. When the depth of the water exceeded a certain amount, the moment on the door caused by the hydrostatic force opened the door. Multiple weights were tested and the depth at which the door opened was measured. This was compared to a theoretically derived model of the system. The comparison showed that the actual depths were greater than the theoretical depths by approximately two centimeters. This may have been due to friction in the system which was not taken into account, as well as a low pressure around the inside of the door created by water leaking through the seal around the door. These components created a moment opposing the opening moment on the door, resulting in a greater depth of water being required to open the door. Introduction / Background The purpose of this laboratory exercise was to experimentally determine the hydrostatic force on a submerged planar surface and compare it to theoretically predicted values. Hydrostatic forces act perpendicular to the faces that a fluid comes into contact with. Due to this, the hydrostatic force caused by the water in Figure 1 will cause a moment on the hinged door pushing outward. The weight, W, will create a tension, T, which will create a moment opposing the moment caused by the hydrostatic force. When the moment caused by the falling weight and the weight of the door, mg, is exceeded by the moment caused by the water, the door will open, releasing the water into the rest of the fish tank. 1 T 2 R L W L’ d mg Water dF s 1 Figure 1: Apparatus used to study the effects of hydrostatic pressure on a submerged surface Assuming that the pulley the weight hangs over is frictionless, the moment caused by the falling weight can be calculated as MW RW sin 2 (1) Similarly, the moment caused by the weight of the door, assuming a uniform density throughout the door, can be calculated as Mmg 0.5L' mg sin1 (2) 2 Taking the sum of these two moments, the total moment acting to close the door is given by Mclosin g RW sin 2 0.5L' mg sin1 (3) The moment caused by the water can be determined by integrating the pressure over the area of the door covered by the water: M open r dF (4) where dF is the force due to pressure acting perpendicular to the door and r is the distance from the hinge. The distance r and the force, dF, can be expressed in terms of s, the distance along the door from the bottom of the door opening. r (L s ) dF p dA (5) p g d s cos 1 dA w ds where w is the width of the door which is in contact with the water. Substituting these into the equation for the moment, d / cos1 M open 0 gw L s d s cos1 ds d / cos1 s 3 cos 1 s 2 M open gw L cos1 dLs 3 2 0 L M open gw 0.5d 2 d cos 3 cos 2 (6) 1 1 By equating Mopen and Mclosing the weight required to balance the water for a given depth of water can be calculated. 3 L RW sin 2 0.5L' mg sin 1 gw 0.5d 2 d cos 3 cos 2 1 1 d3w g ( 0.5) m L' sin 1 cos 1 d L cos 1 w 2 2 3 W (7) R cos 1 sin 2 2 This can be solved for the mass of the weight by dividing both sides by g. d3 w ( 0.5) m L' sin1 cos 1 d L cos1 w 2 2 3 mweight (8) R cos 1 sin 2 2 Using this relationship, a theoretical model can be developed for the behavior of the system which can be compared to actual experimental results. Procedure To perform this experiment, measurements were taken of all quantities from Figure 1 which were not given. A weight was then selected and hung from the string, and water was allowed to flow into the fish tank on the left side. The depth, d, was recorded when the depth was sufficient to open the door. This was repeated for several different weights. Results The quantities given and measured based on Figure 1 are shown in Table 1. Measurements were taken of the depth the water reached before the door opened for various weights. These are displayed in Table 2. 4 Table 1: Measured and given values of lengths, weights, and angles L' 0.165 m L 0.143 m Given w 0.104 m m 0.1495 kg θ1 0.157 rad R 0.114 m Measured θ2 0.611 rad Table 2: Measured water depth based on a hanging mass Depth (m) Mass (g) +/- 0.0005 20 0.043 50 0.047 200 0.063 400 0.086 500 0.093 Substituting values into Equation (8), a relationship is established as shown in the sample calculation below which can be graphed. kg d 3 999 3 0.104m 0.1495kg 0.165m sin0.157 cos 2 0.157 d 2 0.143m 999 kg cos 0.104m ( 0.5) m 1 m3 3 mweight 0.114m cos 0.157 sin0.611 2 kg 3 kg mweight 271.5 3 d 115.024 2 d 2 0.02951kg m m Figure 2 shows the theoretical and actual depth of the water based on the mass of the weight hung from the string attached to the door. 5 0.1 0.09 0.08 0.07 0.06 Depth (m) 0.05 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Mass (kg) Measured Theoretical Figure 2: Plot of measured and theoretical values of depth vs. mass of weight The experimental error between the actual and theoretical values can be calculated as Theoretical actual (9) Error Theoretical Using Equation (9), the experimental error between measured and actual values was calculated. This is shown in Table 3. Table 3: Theoretical and actual values of depth for a given mass, and discrepancies between them Depth (m) +/- Theoretical Depth Discrepancy Experimental Mass (kg) 0.001 (m) (m) Error 0.0200 0.043 0.0213 0.0217 1.02 0.0500 0.047 0.0272 0.0198 0.73 0.2000 0.063 0.0474 0.0156 0.33 0.4000 0.086 0.0666 0.0194 0.29 0.5000 0.093 0.0748 0.0182 0.24 Average: 0.0189 0.52 6 Interpretation and Conclusion The data collected appears to follow a trend similar to the theoretically derived equation. However, the actual data appears to be shifted approximately 0.02m above the theoretical model. This may be partially the result of the friction in the hinges and the pulley, which was not taken into account. This friction would increase the closing moment, requiring the opening moment to be larger to open the door. To make the opening moment larger, the depth of water would have to increase also. Another contributing factor to the actual depth being higher than the theoretical is that water was constantly leaking around the seal on the door. This velocity of the water created a lower pressure around the seal, increasing the closing moment on the door, and again requiring a greater depth of water to open the door. The theoretical model showed that the relationship between the mass of the weight and the depth of the water is a cubic one. This is a logical conclusion, since as the water depth increases, the volume of water (a cubic value) is increasing, and it is this volume which is applying the pressure over the area. 7

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Hydrostatic pressure, the force, hydrostatic force, the fluid, center of pressure, center of gravity, Figure 1, plane surface, balance arm, Free surface

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posted: | 7/2/2011 |

language: | English |

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