Center of Pressure

Hydrostatic Pressure (Center Of Pressure) Objective: The purpose of this experiment is to experimentally locate the center of  pressure of a vertical, submerged, plane surface. The experimental measurement is compared with a theoretical prediction.

Apparatus:

Hydraulic bench, Hydrostatic pressure apparatus, set of weights, metric ruler, graduated cylinders

Theory:

The center of pressure of an immersed body is defined by the vertical distance below the liquid surface. In this experiment we aimed to find out

the center of pressure both theoretically and practically and to compare them after all. Theoretically, calculations of center of pressure " Hcp" by using the Hcp= Hcg + I / Hcg x A Hcg: the center of gravity A: the cross sectional area I: the moment of inertia Hydrostatic force of the fluid pressure acting on the rectangular plane surface. F = ρ x g x A x Hcg Where; ρ: the mass density of water (1000 kg/m3 ) g: the gravitational acceleration (m/s2) A: the cross sectional area of the surface at which the load is acting. Hcg: the center of gravity of the cross section By taking ΣM about the pivot point.

The application of this formula may differ according to whether the body is partially or totally immersed. Practically, the center of pressure will be calculated using the principle of  moment equilibrium of the used apparatus about a pivot, only two forces

create a moment about this pivot as the apparatus is designed to have a curved surface with the pivot as the center, so that forces exerted on this surface all pass through the pivot and create no moment about it. Partial Immersion: F = ρ x g x A x Hcg = ρ x g x y^2/2 x b Hcp = Hcg + I / Hcg x A = 2/3 x y Taking moment around the pivot; m.g.L= F x Hcp = ρ x g x y^2/2 x b [a + d – 1/3 x y] Total Immersion: F= ρ x g x (y-d/2) x (b x d) Hcp= (y – ½) + (d^2 / 12 (y– d/2)) Taking the moment around the pivot; m.g.L= F x Hcp = ρ x g x (y – d/2) x (b x d) x [a + d/2 + d^2/12 x (y – d/2)] Several values of Hcp is talking by mass increments of 50g each trial, and increasing the water level until the hydrostatic force is sufficient to make the apparatus in equilibrium.

Procedures: 1. The instruments used was adjusted as the tank was filled with water till it touched the bottom surface of the surface and that point was considered to be the zero of the vernier caliper. 2. Masses were added to the balance pan in increments of about 50g, and the water surface was raised in the tank until the balance arm is horizontal again. 3. The vernier reading was taken to measure the depth of immersion, which restores the balance arm to its balanced position. 4. A series of readings with increasing values of "m" are then taken.

Analysis and Calculation: Data: result #

Mass (kg)

depth (m)

m.g.L (M1)

F.ycp (M2) 0.15595 5

1

0.05

0.048

0.13488 8

2

0.1

0.064

0.26977 5

0.26921 8

3

0.15

0.082

0.40466 3

0.42710 7

4

0.2

0.096

0.53955

0.56957 6

5

0.25

0.108

6

0.3

0.12

7

0.35

0.134

8

0.4

0.147

1.0791

9

0.45

0.159

1.21398 8

0.67443 8 0.80932 5 0.94421 3

0.64030 9 0.77283 8 0.92747 8 1.07109 3 1.20368

error 0.0210 7 0.0005 57 0.0224 4 0.0300 3 0.0341 29 0.0364 87 0.0167 35 0.0080 07 0.0103 08

Sample of calculations: m.g.L = 0.048*9.81*0.275 = 0.134888 F.Hcp = 9810*(0.134888^2/2)*0.075*(0.1+(0.1-(1/3*0.134888))) = 0.155955

Discussion and Conclusion: The obtained results showed large discrepancies between the theoretical and experimental values of the center of pressure, where the experimental ones were larger than the theoretical ones. The difference between the two values were smaller in the total immersion region (y >= 10cm) than in the partial immersion region. These discrepancies might be a result of errors occurred in the experimental procedures or apparatus. some of the possible errors that might have caused the large discrepancies: 1. Neglecting the weights of the balance and the pan. 2. Errors in determining the depth "y", either due to errors in taking the reading from the vernier or from parallax errors in determining the touching point between the water surface and the pin of the measuring device.

Reference: B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics, 4th ed., 2002, Wiley and Sons, New York.