  # Experiment on the Magnitude of Hydrostatic Forces and the Centre of Pressure in a Submerged Plane Surface

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## Introduction

Hydrostatic Pressure

Hydrostatic pressure is a pressure exerted by fluid from any directions at equilibrium with the presence of gravitational force. Based on this concept,it is learnt that the pressure increases proportionally with the depth/height of the fluid as the weight of fluid exerted downwards increases. The formula for hydrostatic pressure: P=hρg

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Where, P = pressure (Pa)

• = density of liquid (kg/) h = height from object to the surface of the liquid (m) g = force of gravity that is pulling in a downward direction.

### Buoyant Force

Buoyant force is an upward force acting on an object when it is immersed in a fluid. It is discovered and explained in the Archimedes’ principle. according to this principle,the upward force acting on the object is equal to the weight of fluid displaced when the object is partially or fully submerged in a fluid. Center of PressureHydrostatic pressure acts in many directions on a plane surface such in this experiment.

However,the sum of the pressure exerted on the plane surface or the resultant forces will only act through a point is called the center of pressure. The total pressure acting at the center pressure is the integrated vectorial pressure field. Experiment Referring to the figure 3 above, we can obtain the formula to determine the centre of pressure of a partially immersed object by deriving the formula based on the experiment conducted. We can determine that: W = mg……. . (equation 1) From equation 1, we take the weight of the object as W measured in newton. The weight is actually the mass of the object which is influenced by the pull of gravity. Newton’s second law states that if the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force. Hence, we get the equation: F = ma……. . (equation 2) Since the density of a fluid is define as the mass over the volume of the fluid, m = V……. . (equation 3) Thus, by substituting equation 3 into equation 2, we obtain the equation of force in a fluid. F = Vg……. . (equation 4)

According to equation 4, we take F as the force, as the density of the fluid, g as the force of gravity pulling in a downward direction and V as the volume. Based on the experiment, we can display equation 2 in a different way as we replace V with A. F = gA …. . (equation 5) Besides, the quadrant in the experiment is acted by the moment of the weight which is balanced by the moment of the hydrostatic force. The moment of the hydrostatic force is in the opposite direction to the moment of the weight. By taking the moment into calculation, M = W x L……(equation 6) Based on equation 6, W is the force of the moment while L is the radius of the pivot. Moment is the sense of rotation which brings a vector r ( perpendicular distance between the force and the origin ) in line the vector F (force ). W x L = F x r…. . (equation 7) Referring to the experiment, let r = a + d – y + Substituting equation 1, equation 4 and the value of r = a + d – y + into equation 7 mgL = gA ( a + d – y + )………(equation 8) Where W = weight of object, N m = mass of object, kg g = gravity (9. 81 ms-2) F = hydrostatic force, N [image: ] = density of water (1000 kg/m3) A = cross sectional area of surface where force acted on, m2 Hcg = centre of gravity of the cross section, m Hcp = centre of pressure of the cross section, m

According to the theory of centre pressure, the centre of gravity is located at half of the height of an object while the centre of pressure is two third of the height of the object if it is measured from the tip of the surface of the water. Hence, height and height of object from the water surface while the area of immerse surface = b x y: mL = ( by )( y)( a + d – y + y ) mL = ( a + d – y ) (equation 9),equation of partial immersion mL = ( r )…………. (equation 10)

Then, we can derive the equation of r from equation 10 r = (equation 11) Let r become the moment of the arm which is balanced by the moment of the weight, k which is the distance between the pivot line and the surface of water. k = a + d – y………. (equation 12) [image: ] = r – k = r – ( a + d – y )……. . (equation 13) Equation 12 display the method of determining the centre of pressure of a plane submerged in a fluid. By arranging equation 9 which is the equation of partial immersion, we can compare it with the linear equation which is y = mx + c Hence, we obtain the equation: m/ a + d )………(equation 14) Therefore, the gradient of the graph, m is – and the y-intercept is ( a + d )

### Applications

The concept of Hydrostatic pressure has been discovered many decades ago. With the concept applied in the modern era,it has helped us a lot in order to develop and produce more useful products which is widely used in the modern industries might it be in transportation,business and many more. In the food process industry,the concept is formally knowned as High hydrostatic pressure(HHP). Basically,this concept is applied to preserve foods,extending the products shelf life and increase the microbiological safety for the consumption of people. This is crucial as many food manufacturers are not only provide the food supplies locally,but it is going to be consumed by people all around the world.

Other than that,the production of ships and submarines use this concept as they are dealing with liquid.

For instance,a submarine dives up to almost a thousand feet under the sea water for a long time. This causes the submarine is exerted with high amount of pressure. To ensure the safety of the people and the strength of the submarine wall,this is where concept of hydrostatic pressure comes in. The manufacturer has to ensure the proper measurement to be made whilst the cost of production is also to be considered as well. The y-intercept of the graph in this experiment is the area density. Using the theoretical equation and experimental equation, the difference in the value is only 0. 05. Since the value of the y-intercept difference is 0. 05, it is considered that the difference is very small. Besides, both the gradient and the y-intercept are different when we compare the theoretical and experimental values. There are a lot of reasons why the discrepancies exist.

Some of them are:

• Parallax error which is due to the error when taking the reading of the height of the water level. Besides, parallax error also occurs during the experiment when balancing the balance arm to make it horizontal so that it will align with its initial point.
• Besides, the water used in this experiment is tap water. Hence, the density of the water is not exactly 1000 kg/ since it contains a lot of impurities and the presence of air bubbles might affect the volume of water used in the experiment.
• The movement of the wind also influenced the result of the experiment. Hence, there are some precautions that can be taken into account to reduce some of the error.
• To reduce parallex error, the eye of the observer should be perpendicular to the measurement to obtain an accurate reading.
• Besides, we should ensure that the balance pan is not moving when taking the reading of the measurement.
• We should use distilled water when conducting the experiment so that the density of the fluid is 1000 kg/m3. Besides, it also does not contain any impurities that could influence the experiment results.
• The experiment should be done in a lab that is inert so that there will be no wind factor that could affect the experiment.

## Conclusion

This experiment was conducted mainly to determine the centre pressure of a plane when it is partially and fully submerged in water. The value of centre pressure of the plane based on the theory and experiment has been found. We also obtained the theoretical equation, and the experimental equation, which is obtained based on the experiment results and graph plotted. the difference in the value obtained in the experiment with the theoretical value might be due to the inaccurate apparatus used and error done while taking the reading such as the parallex error. The water used in the experiment might also contain some other impurities and air bubbles which affect the real density and volume of the water.

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