Hydrostatic Lab experiment PDF

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An experiment was conducted to test the use of an Essom apparatus for determining the center of pressure of a hydrostatic force acting on a submerged vertical surface. The following two cases were examined in the experiment: fully submerged case and partially submerged case. For each case, nine ru...

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Hydrostatic Force and Center of Pressure on a Vertical Submerged Surface

Abstract An experiment was conducted to test the use of an Essom apparatus for determining the center of pressure of a hydrostatic force acting on a submerged vertical surface. The following two cases were examined in the experiment: fully submerged case and partially submerged case. For each case, nine runs of measured mass and height readings were obtained. Two plots were done for each case: YR vs. h and FR vs. h. As expected, the percent error was very small for the fully submerged case but grew substantially for the partially submerged case. This is discussed in detail to follow.

Introduction When an object is submerged in a fluid, the fluid imparts forces on the object’s surfaces. When a fluid is at rest and no shearing forces exist, the force exerted by the fluid is perpendicular to a plane surface [2]. This principle was explored through the completion of the following objectives: 1. Determine the center of pressure (yR = hR) of a hydrostatic force acting on a vertical plane surface immersed in water both experimentally and theoretically. 2. Determine the validity of the experimental results based on the theoretical results.

According to Al Masoud [1], for the fully submerged case the magnitude of the force (FR) and the experimental depth of center of pressure (yRE) are respectively as follows: F R=γ h c A y ℜ=d− ( 0.2−h )=

mL −(0.2−h) ρ ( h−0.05 ) hw

(1) (2)

Similarly, the theoretical line of action of the hydrostatic force for the fully submerged case is calculated as follows [2]: y RT =

Ic +y yc A c

(3)

Experimentally, the center of pressure yRE (m) depends on the water depth h (m) and the distance, d (m), between the line of action and the balance beam. It should be noted here that d is a function of the water depth and mass as seen above. Theoretically, the center of pressure yRT (m) depends on the second moment of area with respect to an axis passing through the centroid (m4) and the distance from the free surface to the center of gravity yc (m). For the fully submerged case, only hc = yc is a function of the fluid height, h, since the area of the submerged plane surface remains constant. While the derivation of the equations used for the partially submerged case is similar to that of the fully submerged case, several differences exist. In this case, the submerged area, also known as the “wet area,” is also function of the fluid height in addition to hc. Experimentally, the line of action of the hydrostatic force is calculated as follows: y ℜ=d− ( 0.2−h )=

2 mL −( 0.2−h) ρ h2 w

(4)

The theoretical center of pressure for the partially submerged case is calculated as follows: 2 y RT = h 3

(5)

The differences between these equations and those used for the fully submerged case can be explained. For the fully submerged case, the hydrostatic force varies linearly with the water height since the wet area remains constant. For the partially submerged case, however, the area is a function of the fluid depth. This causes the hydrostatic force to have a quadratic relationship with the water depth as opposed to a linear one. In both cases, however, the moments are summed about the fulcrum in the derivation of the center of pressure equations. Finally, the percent error for both the fully submerged case and the partially submerged case is calculated using the following equation:

|y ℜ− y RT|

% Error =

y RT

∗100

(6)

Methods and Materials The Essom apparatus used in the experiment consisted of a Plexiglas box, a quadrant mounted on a balance arm, and various mass blocks used to balance the apparatus. The quadrant pivots and the pivot point is coincident with the quadrant’s axis. In other words, the only hydrostatic force that needs to be considered is the force on the vertical face since all other forces

cancel out or pass through the point of rotation [1]. A photograph and the technical specs of the Essom apparatus used are seen below in Figure 1 and Table 1 respectively.

Figure 1: Essom Apparatus

Table 1: Technical Data of Essom Apparatus

Description Weight hanger to fulcrum, L Bottom of quadrant vertical face to pivot, Do Top of quadrant vertical face to pivot, Di Height of quadrant vertical face Width of quadrant vertical face, W

Distance (m) 0.280 0.200 0.100 0.100 0.075

To better help visualize the forces being considered during this experiment, Figure 2 and Figure 3 depict the free body diagrams of the apparatus for the fully submerged case and the partially submerged case, respectively.

Figure 2: FBD of the Fully Submerged Case

Figure 3: FBD of the Partially Submerged Case

The following procedure was used to obtain the basic experimental data:

1. 2. 3. 4. 5.

Position the empty tank on a working surface. Adjust the screwed feet until the apparatus is level. Move the counter-balance weight until the balance arm is horizontal. Make sure the drain valve is closed and fill the tank up with water. Add weights to the hanger to retain the balance of beam. Water can be added or drained from the tank in order to achieve equilibrium. 6. Read the depth of immersion from the scale (reading should be taken from the meniscus and the line of sight should be slightly below the water surface). 7. Remove the weights and drain some water for the next, new equilibrium state. 8. Repeat Steps 5-7 for both the fully submerged case and the partially submerged case until nine runs total have been logged for each case.

Results The basic data recorded during the experiment for the fully submerged case and the partially submerged case are shown in the first three columns of Table 2 and Table 3 below. Furthermore, these tables contain the calculations for the experimental considerations, the theoretical values, and the percent error. Table 2: Experimental and Theoretical Results for Fully Submerged Case

Run

Mass (kg)

h (m)

1 2 3 4 5 6 7 8 9

0.61 0.56 0.53 0.49 0.43 0.42 0.40 0.33 0.32

0.198 0.186 0.178 0.169 0.154 0.151 0.146 0.128 0.126

Fully Submerged Experimental hc (m) 0.148 0.136 0.128 0.119 0.104 0.101 0.096 0.078 0.076

FR (N) 10.878 9.996 9.408 8.747 7.644 7.424 7.056 5.733 5.586

d (m) 0.1540 0.1539 0.1547 0.1539 0.1545 0.1554 0.1557 0.1581 0.1574

yRE (m) 0.1520 0.1399 0.1327 0.1229 0.1085 0.1064 0.1017 0.0861 0.0834

yRT (m) 0.1536 0.1421 0.1345 0.1260 0.1120 0.1093 0.1047 0.0887 0.0870

% Error 1.04 1.58 1.32 2.48 3.12 2.60 2.83 2.90 4.15

Table 3: Experimental and Theoretical Results for Partially Submerged Case

Run

Mass (kg)

h (m)

1 2 3 4 5

0.20 0.17 0.16 0.15 0.14

0.098 0.090 0.087 0.084 0.080

Partially Submerged Experimental hc (m) FR (N) d (m)

yRE (m)

0.0490 0.0450 0.0435 0.0420 0.0400

0.0536 0.0469 0.0450 0.0429 0.0435

3.5295 2.9768 2.7816 2.5931 2.3520

0.1556 0.1569 0.1580 0.1589 0.1635

yRT (m) 0.0653 0.0600 0.0580 0.0560 0.0533

% Error 17.88 21.89 22.42 23.41 18.44

6 7 8 9

0.12 0.10 0.08 0.05

0.075 0.068 0.060 0.044

0.0375 0.0340 0.0300 0.0220

2.0672 1.6993 1.3230 0.7115

0.1595 0.1616 0.1661 0.1930

0.0345 0.0296 0.0261 0.0370

0.0500 0.0453 0.0400 0.0293

31.10 34.62 34.76 26.25

The data in Table 2 and Table 3 is plotted in the charts below. Figure 4 is a plot of yR versus the depth of immersion, h, for both cases. The chart shows both the experimental and theoretical values for the line of action of the hydrostatic force. Figure 5 is a plot of the hydrostatic force versus the depth of immersion. Finally, Figure 6 shows how the percent error is influenced by the height of the water for both cases.

yR vs. h h (partially submerged) (m) 0.1800 0.1600 0.1400

f(x) = − 0.01 x + 0.16

yR (m)

0.1200 0.1000 0.0800 0.0600 0.0400

f(x) = − 0 x + 0.05

0.0200 0.0000 0.098 0.098

0.090 0.090

0.087 0.087

0.084 0.084

0.080 0.080

0.075 0.075

0.068 0.068

0.060 0.060

0.044 0.044

h (fully submerged) (m) Fully, Experimental Fully, Theoretical Linear (Partially, Experimental)

Linear (Fully, Experimental) Partially, Experimental Partially, Theoretical

Figure 4: Graph of yR vs. h for Fully Submerged and Partially Submerged Cases

f(x) = − 0.32 x + 3.8

FR vs. h h (partially submerged) (m) 12.000

10.000

f(x) = − 0.67 x + 11.38

6.000

4.000

2.000

0.000 0.098 0.098

0.090 0.090

0.087 0.087

0.084 0.084

0.080 0.080

0.075 0.075

0.068 0.068

0.060 0.060

0.044 0.044

h (fully submerged) (m) Fully, Experimental Partially, Experimental

Linear (Fully, Experimental) Linear (Partially, Experimental)

Figure 5: Graph of FR vs. h for Fully Submerged and Partially Submerged Cases

Percent Error vs. h h (partially submerged) (m) 5 4 4

Percent Error (%)

FR (m)

8.000

3 3 2 2 1 1 0

0.098 0.098 0.090 0.090 0.087 0.087 0.084 0.084 0.080 0.080 0.075 0.075 0.068 0.068 0.060 0.060 0.044 0.044 h (fully submerged) (m) Fully, Experimental

Partially, Experimental

Figure 6: Percent Error for Fully Submerged and Partially Submerged Cases

Discussion As seen in Table 2 and Table 3, the percent error differs greatly between the fully submerged case and the partially submerged case. For the fully submerged case, all of the errors for the individual runs remained below 5% with the largest being 4.15%. In general, the percent error increased as the amount of water in the tank (indicated by h in the table) decreased. This trend carried over to the partially submerged case. For this case, the percent errors increased dramatically ranging from 17.88% to 34.76%. Lab groups were notified of this prior to completing the lab so this was expected. Furthermore, both YR and FR decreased in magnitude as the water level dropped. While this was seen in both cases, the graphs in the appendix show that the fully submerged case declined at a faster rate (in terms of both YR and FR) than the partially submerged case. It makes sense that both YR and FR would decrease with the volume, or height, of the water. This is because YR is measured from the free surface of the liquid. In addition, as water is removed from the tank, less of a hydrostatic force is required to balance the apparatus. In terms of suggestions for the experiment, it would be interesting to conduct the experiment with both apparatuses (the Essom apparatus and the Armfield apparatus). The percent errors of each apparatus could then be compared. In addition, it would be interesting to conduct an additional section of the lab where an apparatus with a non-vertical surface is studied. For example, if the same type of apparatus was used with the only difference being that the hydrostatic force was examined to be acting on a surface inclined at some certain angle. Such cases were studied and class and would be valid to study experimentally.

Conclusions The following conclusions can be made:     

hc = yc and hR = yR for a hydrostatic force acting on a vertical surface The largest percent error for YR in the fully submerged case is 4.15% The largest percent error for YR in the partially submerged case is 34.76% o The percent error for the partially submerged case is substantially greater than that of the fully submerged case yR decreases as the volume, measured in terms of height, decreases o yR decreases at a faster rate for the fully submerged case FR decreases as the volume, measured in terms of height, decreases o FR decreases at a faster rate for the fully submerged case...