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Hydrostatic Forces

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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 sin1                               (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 sin1                     (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 / cos1
                M open  
                            0
                                         gw L  s d  s cos1 ds
                                                                            d / cos1 
                                  s 3 cos 1  s 2                    
                M open    gw                   L cos1   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 

                                                                          d3w 
                  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' sin1 cos 1   d L cos1 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  sin0.157  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  sin0.611
                                                                            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.




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