Sem 3 - Lab Metacentric Apparatus [PDF]

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[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



INTRODUCTION The Metacentric Height Apparatus enable student to vary the position of the metacentre to produce stable and unstable equilibrium4. The equipment consists of a plastic rectangular floating pontoon, the centre of gravity which ca be varied by an adjustable weight which slides can be clamped in any position on a vertical mast1. A single plumb bob is suspended from the mast which indicates the angle of heel on a calibrated scale2. A weight with lateral adjustment allows the degree of heel to be varied and hence the stability of the pontoon determined2. The equipment does not require a separate water tank as it may be used on the Hydraulics bench by filling the volumetric tank2.



The stability gradient can be calculated as follows :



Where



While x, α are position and angle values measured from apparatus1



The aim of this experiment s to study the stability of a floating body experimentally and compare with the calculated stability1



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[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



THEORETICAL BACKGROUND For a body to be equilibrium W=Fb and both weight of the body (W) and the buoyant force (Fb) are acting along the same vertical line for a body in equilibrium in liquid surface the two forces gravity force (w) and buoyant force (Fb) must lie in the same vertical line and for stable equilibrium point M must be above G.



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[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



The position of the Metacentre relative to the position of the centre of gravity of a floating body determined the stability of the floating body. 1. Stable equilibrium: If the point M is above G, the floating body will be in stable equilibrium. 2. Unstable equilibrium: If the point M is below G, the floating body will be in unstable equilibrium. 3. Neutral equilibrium: If the point M is at the centre of the gravity of the body, the floating body will be in neutral equilibrium. Consider a ship or pontoon floating as shown in figure 2. The centre of gravity of the body is at and the centre of buoyancy is at B. For equilibrium, the weight of the floating body is equal to the weight of the liquid it displaces and the centre of gravity of the body and the centroid of the displaced liquid are in the same vertical line. The centroid of the displaced liquid is called the "centre of buoyancy". Let the body now be heeled through an angle θ, B1 will be the position of the centre of buoyancy after heeling. A vertical line through B1 will intersect the centre line of the body at M and this point is known as the metacentre of the body when an angle θ is diminishingly small. The distance GM is known as the metacentric height. The force due to buoyancy acts vertically up through B1 and is equal to W. The weight of the body acts downwards through G.



Figure : Illustrative figure of flat bottomed pontoon



Figure : Centers buoyancy of floating and submerged objects.



Stability of submerged objects: Stable equilibrium: if when displaced, it returns to equilibrium position. If the centre of gravity is below the centre of buoyancy, a righting moment will have produced, and the body will tend to return to its equilibrium position (Stable). Unstable equilibrium: if when displaced it returns to a new equilibrium position. If the Centre of Gravity is above the centre of buoyancy, an overturning moment is produced, and the body is (unstable). Note: As the body is totally submerged, the shape of displaced fluid is not altered when the body is tilted and so the centre of buoyancy unchanged relative to the body.



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[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



EXPERIMENTAL SETUP The setup consists of a small water tank having transparent side walls in which a small ship model is floated, the weight of the model can be changed by adding or removing weights. Adjustable mass is used for tilting the ship; plump line is attached to the mast to measure the tilting angle



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[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



PROCEDURE 1. The weight positions (x and z) is set to the apparatus referring to the figure. 2. Vertical sliding weight is moved to bottom position. 3. Tank is filled with provided water and floating body is inserted. 4. Vertical sliding weight is raised gradually and angle on hell indicator is read off. Height of sliding weight at top edge of weight and enter in table together with angle is read off.



RESULTS Table of various z at constant x values. x 10 10 10 15 15 15



z 5 10 15 5 10 15



Angle ( °) 2.0° 5.0° 14.0° 3.0° 5.0° 17.0°



Stability Gradient 0.275 0.11 0.039 0.275 0.165 0.049



Table of various x at constant z values. x 5 10 15 5 10 15



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z 10 10 10 15 15 15



Angle ( °) 3.0° 5.0° 7.0° 10.5° 14.0° 17.0°



Stability Gradient 0.275 0.11 0.039 0.275 0.165 0.049



[LAB EXPERIMENT: METACENTRIC HEIGHT APPARATUS]



September 25, 2018



DISCUSSION Metacentre is the point which the body start oscillating whenever the floating body in a liquid given a small angular displacement. In fluid mechanics, metacentre is a theoretical point which an imaginary vertical line passing through the centre of gravity and centre of buoyancy. Based on the experiment, the position of the metacentre does depend on the position of the centre of gravity. This is because when the centre of gravity change, the intersection of the imaginary vertical line also changes. Thus, the position of the metacentre change. Metacentre height is a measurement of the initial static stability of a floating body. It can be obtained from the distance between the metacentre and centre of gravity of the body. Metacentre height is important to stability. It does vary with angle of heel. The angle of heel depends on stability. The lower of metacentre height, the more stable of the body so the angle of heel is reduced. Changing the position of gravity will affect the position of the metacentre. This is because changing the centre of body will create a new imaginary vertical line, and a new intersection with imaginary vertical line of centre of buoyancy. Thus, the position of metacentre will change too. The values of metacentre height, GM at lowest level of theta, θ are likely to be less accurate because at lower range level of theta is seemly giving the reading of the metacentre height just a little different. The GM also does not have high accuracy of reading which cannot provide accurate reading. Besides, the body will right itself more forcefully when near to lowest theta. This movement of body also spoil the reading obtained.



CONCLUSION In conclusion, the stability gradient of the floating body is inversely proportional to the angle of heel, α which mean the smaller the angle of heel, the grater the stability gradient of the floating body. This experiment helps to study to make floating object more stable which is applicable to the ship, sailboat or other floating object and improve the application in future.



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REFERENCE



1. [website] Metacentric height. (2018, May 14). Retrieved from https://en.wikipedia.org/wiki/Metacentric_height 2. [website] A., Grigoriadis, C. F., Shaik, F., Raghul, G., & Michael, A. A. (2016, December 26). Inclining Experiment- Determining Metacentric height of the ship. Retrieved from https://www.marineinsight.com/naval-architecture/inclining-experiment-determiningmetacentric-height-of-the-ship/ 3. [website] Fluid Mechanics Applications/A:20 Stability and oscillation of floating bodies. (n.d.). Retrieved from https://en.wikibooks.org/wiki/Fluid_Mechanics_Applications/A:20_Stability_and_oscillation _of_floating_bodies 4. [book] Patra, K. (2011). Engineering fluid mechanics and hydraulic machines. Oxford: Alpha Science International. 5. [book] White, F. M. (2017). Fluid mechanics. New Delhi, India: McGraw-Hill Education. 6. [book] Hibbeler, R. C. (2018). Fluid mechanics. NY, NY: Pearson.



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