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MM 222 Fluid Mechanics

The University of the South Pacific School of Engineering and Physics MM 222 – Fluid Mechanics Lab 1: Hydrostatic Force and Centre of Pressure on a Submerged Surface Lab Session Time: Thursday (5 to 8pm) Student Name: Sahil Sharma; Sanya Gibson; Adi Deborah Raiwalui Student ID#: S11171799; S11160329; S11175429 AIM This experiment aims to determine the hydrostatic forces acting on the plane surface submerged in water and the depth of center of pressure at which said forces are acting. Furthermore, this experiment also aims to compare theoretical values with experimental values to see how much they differ.

INTRODUCTION For clarity,” hydrostatics” is the term given to describe fluids (either liquids or gases) that are at rest. In this state, hydrostatic pressure induced is defined as the pressure exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity [ CITATION EDi20 \l 3081 ]. Hence, as the depth of the fluid measured from the free surface increases, the hydrostatic pressure also increases. Consequently, with the presence of hydrostatic pressure comes hydrostatic force. Acting on the plane surface of the submerged body, hydrostatic forces form a system of parallel forces, whereby the magnitude of the force is calculated in addition to its point of application known as the depth of center of pressure [CITATION Yun14 \l 3081 ]. The depth of center of pressure is denoted by “yp”. Understanding the significance of these concepts is vital for many engineering applications (Munson and Young, 2009) determining these forces is important in designing storage tanks, ships, dams and other hydraulic structures. With the experimental setup shown below, this lab report is being conducted with the intention of comparing experimental values with the results we will have obtained from theoretical formulas. Ultimately, the lab report serves to help students gain a better grasp of the subject of hydrostatics and its behavior.

Figure 1: Experimental setup for determining hydrostatic pressure Source: School of Engineering and Physics [CITATION The20 \p 1 \n \l 3081 ]

Lab Report

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MM 222 Fluid Mechanics EQUIPMENT The following is a list of the types of equipment and materials that would be used in the hardware experiment.  

Hydraulic bench Hydrostatic Pressure Apparatus

METHODOLOGY This procedure was conducted in the laboratory under supervision of the lab instructor. The materials that were used to conduct this experiment were Hydraulic bench, Hydrostatic Pressure Apparatus (fig 1) and counter weights. The value of the depth of immersion was recorded along with the counter weight that took to balance the apparatus. The steps to obtain these data are as follows: 1.) The tank of the Hydrostatic pressure was filled with water to 100mm mark of the quadrant located in the tank. 2.) Counter weights were then added on to balance out the arm until it was in middle of the beam level indicator. 3.) The value of ‘r’ (depth of immersion) were read and recorded from the scale and the counter mass used to balance the arm was also recorded 4.) More water was poured into the Hydrostatic Pressure tank and was counter balanced to obtain a new value of ‘r’ (depth of immersion) 5.) This procedure was again repeated 2 more times and each new value of ‘r’ and mass were recorded and tabled.

RESULTS AND DISCUSSION Table 1 Hydrostatic force at different depth of immersion Theoretical (mm)

Experimental (mm)

Force

yn

yp

yn

yp

N

52.5

165

70

156.62

61.6

4.05

0.260

56.5

162.33

75.33

152

65

4.69

84

0.280

58

161.33

77.33

161.33

69

4.94

80

0.2925

60

160

80

149.26

69.25

5.29

r (mm)

q (mm)

m (kg)

yc (mm)

105

95

0.235

113

87

116 120

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MM 222 Fluid Mechanics Sample Calculation for (r =105mm) *Constants p=275mm b=75mm d=100mm

q = 200 – r = 95mm

ρwater = 998 kg/m3

yc =

Area = L x w = 75 x 100 = 7500mm2

r 2 105 = 2 =52.5mm

Theoretical Values

yp theoretical =

2 r 3 2 (105) = 3 = 70mm

yn theoretical = yp theoretical + q =70 + 95 =165mm

Experimental Values yn experimental =

2 Pm ρb r 2 2(275 )( 235) = (998 )(75 )(105 )2 = 156.62mm

yp experimental = yn experimental - q = 156.62 - 95 = 61.62mm

* force acting on the plane surface 1 ρgb r 2 2 = (0.5)(998)(9.81)(0.075)(0.105)2 = 4.05N

F=

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MM 222 Fluid Mechanics

Force (N)

Hydrostatic Force V Immersion Depth 6 5 4 3 2 1 0 104 106 108 110 112 114 116 118 120 122

Immmersion Depth(mm)

Center of Pressure(mm)

Figure 2: Hydrostatic Force V Immersion Depth Source: MS Excel

Center of Pressure v Immersion Depth 70 68 66 64 62 60 58 56 104 106 108 110 112 114 116 118 120 122

Immersion Depth(mm) Figure 3: Center of Pressure v Immersion Depth Source: MS Excel

“For fluids at rest we know that the force must be perpendicular to the surface since there are no shearing stresses present (Munson and Young, 2009). We also know that the pressure will vary linearly with depth…” In other words, since the pressure increases with depth, it is therefore only logical to assume that the force will also increase. Thus, the graph shown in Figure (2) is significant as it demonstrates the proportionality between hydrostatic force and depth of immersion appropriately. Figure (3) establishes that depth of center of pressure increases as depth of immersion increases. Upon speculation, this relationship holds true because as the object continues to be submerged, the hydrostatic forces acting on its vertical plane surface will continue to escalate. As a result, y p will also change to counter this reaction in order to prevent the submerged plane from changing its orientation.

CONCLUSION

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MM 222 Fluid Mechanics After conducting this experiment, the hydrostatic force, pressure and the position where the force acts were determined. The value of the depth of the center of pressure (y p) were compared to the values that were obtained from theoretical formulas. Upon comparison of the values it was observed that the values were close to each other therefore with reference to the a both the graphs above, it can be said that hydrostatic force and center of pressure are directly proportionate to the immersion depth

REFERENCE Cengel, Y. A. & Cimbala, J. M., 2014. Hydrostatic Forces on Submerged Plane Surfaces. In: Fluid MechanicsFundamentals and Applications. New York: McGraw Hill, p. 89. EDinformatics, n.d. What is Hydrostatic Pressure --- Fluid Pressure and Depth. [Online] Available at: https://www.edinformatics.com/math_science/hydrostatic_pressure.htm#:~:text=Hydrostatic %20pressure%20is%20the%20pressure,exerting%20downward%20force%20from%20above. [Accessed 23 August 2020]. Munson, B. R. & Young, D. F., 2009. Hydrostatic Force on a Plane Surface. In: Fundamentals of Fluid Mechanics [6 ed]. s.l.:John Wiley & Sons, Inc, p. 57. The University of the South Pacific: School of Engineering and Physics, 2020. Lab 1: Hydrostatic Force and Center of Pressure on a Submerged Surface, s.l.: s.n.

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