Lab Report Centre of pressure on a plane Surface

Abstract
Water dams and water gates are of major importance in the aggregation and environmental hazards protection fields. As the amount of water and the water level of a reservoir constantly vary, the pressure force on the side of the dam varies and thus the moment bending the dam as well. The moment does not increase linearly due to the increase in the pressure force itself rather it increases in a different form, because it depends on the level of the water as well. The level of the water affects the force but affects the moment arm as well. Through these pages we shall theorize, experiment, and discuss how does the level of the water affects the force and the moment arm or more generally Pressure force and Center of pressure.

Introduction
Fluid Statics or Hydrostatic A fundamental characteristic of any fluid at rest is that the force exerted on any particle within the fluid is the same in all directions. Or more conveniently a symmetric body when immersed in a fluid and further pushed downwards will only suffer an upward resistance due to the buoyancy force exerted on it but will never move or tilt (rotate) in any direction because a resultant circumferential force is absent although its surfaces are under hydrostatic pressure theorized to increase with depth. On the contrary an asymmetric body when submerged and further pushed down a liquid will be acted upon by a moment resulted from the difference in the forces acting on the sides of the asymmetric body.

Objectives

To determine the hydrostatic thrust acting on a plane surface immersed in water. To determine the position of the line of action of the thrust and to compare the position determined by experiment with the theoretical position.

Method and Equipment

Method: A body shaped as quarter of a ring (a quadrant) is mounted on a balance arm, which pivots on knife edges. The line of contact of the knife edges coincides with the axis of the quadrant. Thus, of the hydrostatic forces acting on the quadrant when immersed not producing moment about the knife edge axis, only the force on the rectangular face end gives rise to a moment about the knife edge axis. In addition to the quadrant clamping screw the balance arm incorporates a balance pan, an adjustable counterbalance and an indicator which shows when the arm is horizontal. The Perspex tank should be levelled by adjusting the screwed feet. Correct alignment is indicated by a circular spirit level mounted on the base of the tank. Water is admitted to the top of the tank by a flexible tube and may be drained through the drain cock in the base. The water supply is obtained from the hydrostatic work bench. The water level in the Perspex tank is indicated on a scale. By achieving an equilibrium condition between the moments acting on the balance arm of the test apparatus. The forces concerned are the weight force W applied to the balance and the hydrostatic pressure thrust F on the end face of the quadrant.

Equipment or Materials:
123-

Weights (masses) with a hook each mass weights 50g, one 25g, one 20g, and one 10g. Clean water. Hydraulic Pressure Apparatus F1-12 shown below.

Technical Data of the f1-12 apparatus: The following dimensions from the equipment are used in the appropriate calculations. These values should be checked as part of the experimental procedure and replaced with the measurements or values found (calibration) since they are provided by the manufacturer. Length of Balance (L) 275 mm Distance from weight hanger to pivot. Quadrant to Pivot (a+d) 200 mm Base of quadrant face to pivot height. Height of Quadrant d 100 mm Height of vertical quadrant face. Width of Quadrant b 75 mm Width of vertical quadrant face.
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Theory

PRESSURE ON A SURFACE IMMERSED IN A LIQUID Because the basic theory for the partly submerged and fully submerged plane is the same, only the partially submerged case will be considered. 1- Partially submerged vertical plane surface

Thrust on surface: Where And Thus ......................... Eqn. 1

Moment of thrust about pivot:

Where = depth of line of action of thrust below pivot. I.e. centre of pressure P. Equilibrium condition: A balancing moment is produced by the weight (W) applied to the hanger at the end of the balance arm For static equilibrium the two moments are equal. i.e. Thus Center of pressure: It is known that pressure relates to depth as (metres) (m = applied mass) ...............................Eqn. 2

Where z = depth of point under consideration measured from the water surface downwards i.e. (-ve) Thus it has a linear distribution with a constant slope as shown below: Geometrically: height of point p (e) is the same as the height of the center of area of the triangle. i.e. (meters)

Thus Substitute Eqn.3 in Eqn.2 to get:

...............................Eqn. 3

From which we can deduce:

( )

...............................Eqn. 4

2- Fully submerged vertical plane surface This will not be taken into account because the previous serves enough to investigate the effects of water level on the center of pressure.

Procedure Steps
Procedure - Equipment Calibration:

Measure the dimensions b and d of the quadrant end-face and the distances (a) and L. Update the values in the results table as necessary.
Procedure - Equipment Set Up:

Position the empty F1-12 tank on a stiff table, and adjust the screwed feet until the built-in circular spirit level indicates that the base is horizontal. Position the balance arm on the knife edges. Locate the weight hangar in the groove at the end of the balance arm. Move the counter-balance weight until the balance arm is horizontal.
Procedure - Taking a Set of Results:

First the hook that the masses will be hooked with weights 50g producing moment inclining the balance arm. Add water until the hydrostatic thrust on the end-face of the quadrant causes the balance arm to rise. Ensure that there is no water spilled on the upper surfaces of the quadrant or the sides, above the water level. Continue to add water until the balance arm is horizontal, measuring this by aligning the base of the balance arm with the top or bottom of the central marking on the balance rest (either can be used, but it must be kept consistent during the experiment). It is easier to slightly over-fill the tank, and obtain the equilibrium position by opening the drain cock to allow a small outflow.

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Read the depth of immersion from the scale on the face of the quadrant; more accurate results can be obtained from reading with the line of sight slightly below the surface, to avoid the effects of surface tension. Add a small mass (50g) or (20g) as load increments to the weight hanger and repeat the above procedure for each load increment, produced by adding a further weight to the weight hanger. The weights supplied allow increments of ten, twenty, and fifty grams to be used, depending on the number of samples required. Fifty-gram intervals mixed with twentyfive-gram intervals are suggested for an initial set of results, which will give a total of six samples. Continue until the water level reaches the top of the upper scale on the quadrant face. Repeat the procedure in reverse, by progressively removing the weights. Note any factors that you think are likely to affect the accuracy of the results.

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RESULTS
m : kg 0.05 0.1 0.12 0.15 0.2 0.22 y : meters X-axis 0.045 0.065 0.071 0.08 0.094 0.1 m/y^2 Y-axis 24.69135802 23.66863905 23.80480063 23.4375 22.63467632 22

Using linear regression analysis yields Slope, m, = y, intercept = Correlation,R = -45.61133494 26.83168857 -0.972863342

Calculations:

] ( )
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Discussion

25 24.5 24 23.5 23 22.5 22 21.5 0 0.02 0.04 0.06 0.08 0.1 0.12

y = -45.61x + 26.83 R = 0.946

Discrepancies

Y experiment 24.69135802 23.66863905 23.80480063 23.4375 22.63467632 22 -0.972863342

Yc curve 24.7791785 23.8669518 23.59328379 23.18278178 22.54422309 22.27055508

|y| 0.087820475 0.198312748 0.211516843 0.254718223 0.090453236 0.270555079

Correlation,R =

range= (|y|)max=

2.691358025 0.270555079 0.100527346 10.05%

e =(|y|)max/range=
y

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The correlation factor R deviates somewhat from unity. Maximum linear error is about ten percent. Discrepancies are mainly due to human error source, surface wetting, and thus surface tension. If measurements were taken up and then down that is to check for hysteresis, Hysteresis would prove to be present even slightly. The reason for hysteresis is that when we take the measurements upward filling the tank adhesive forces or surface tension between the water level and the face of the quadrant are large, because the quadrants face is still dry. But when we take the measurements downward emptying the tank adhesive forces will be smaller because the surface is already wet and the rough texture would be saturated with water.

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Conclusion
The formulas derived in the theory section are nearly accurate if only were some discrepancies not present. If another device was used to further insure the value of the density of the water used as it is not pure water.

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