17

FLUIDS MECHANICS LAB: METACENTRIC HEIGHT

INSTITUTE OF TECHNOLOGY TALLAGHT

Student Name: Son HoangStudent Number: X00067077GROUP MEMBERS: Joe stynes, Danielle warner, noel, Benjamin VaroLecturers: c. Costello, K. MaloneyCOURSE CODE: TA_EMECH_D

OBJECTIVES

The objective of this lab is 1) To determine the metacentric height for a floating pontoon, and compare the theory with the recorded figures. 2) To test Archimedes principle for a body immersed / floating in a fluid, stating that If the weight of an object is heavier than the water displaced it will sink otherwise it will float.

BACKGROUND THEORY Archimedes principle for a body immersed / floating in a fluid:There is an upward buoyancy force equal to the weight of the fluid displaced and acting through the centroid of the displaced fluid. (The floating body displaces its own weight of the liquid in which it floats. The idea is that the floating body should not tip over when the floating barge tips slightly to one side.1. the overall volume of the displaced fluid does not change2. The shape of the displaced fluid does change as the body dips in deeper on one side and lifts out of the fluid on the other side. This means that the position B, of the buoyancy force moves to one side.3. The point where the centreline and the vertical force meet M is called the METACENTER. The distance between M and G (GM) is called the METACENTRIC HEIGHT.4. When the body tips through an angle, the restoring moment is;

(http://accessscience.com/loadBinary.aspx?alD=3672&filename=330900FG0040.gif) The Centre of buoyancy (COB) is a force that acts through the centre of gravity (COG) of a displaced fluid. It has three states of equilibrium as follows.1. Stable: if a force is applied and released to a body, the body moves back to the original position. (COG is below COB)2. Unstable: if a force is applied to a body, the body moves away from the original position. (COG is above COB)3. Neutral: if a force is applied to a body, the body moves accordingly to the magnitude of the force and DOES NOT return to the original position/ continues to move further away.

BACKGROUND THEORY THEORETICAL formulas. The following theoretical formulas for the metacentric height (GM) are as follows:

Where Where = Where and = immersed depth of the floating rig B = centroid of the displaced volume G = centre of gravity of the rig. This is dependent on how the load is distributed. BM = metacentric radius, measure of restoring moment when the rig tilts. Dependant on the shape of the rig. A wide rig gives better stability. GM = metacentric height, the measurement of stability. Dependant on the distance of BM and BG.

BACKGROUND THEORY

EXPERIMENTAL Formulas

Where Using algebra to manipulate the above formula, we get

PROCEDUREEquipment 1) sliding mass

2) Traversing Mass

3) Linear scale

4) Floating pontoon / rig

Procedure:Equipment5) Tub of water(http://www.reptiles1.com/images/empty_tub.jpg)

6) Plumb line

7) Digital weighing scale (http://www.satwikscales.com/images/products/big/weighing_scale_IQ_100/weighing_scale_IQ_100.jpg)

8) Ruler

(http://cdn.dickblick.com/items/554/30/55430-1012-1-3ww-l.jpg)

Procedure

Steps

1) Weigh the traversing mass (P)2) Assemble the pontoon rig with the traversing mass in the mid position and weigh the entire assembly (W)3) Position the sliding mass at a point low down on the mast. Measure the distance from the base of the tank to the centre of gravity (y).

4) Fill the sink with water and float he pontoon, ensuring that the traversing mass is in its central position.5) Check zero datum between plumb line and scale.6) Record any data needed for calculating the THEORETICAL metacentric height7) Move the traversing mass to the right of centre in 20mm increments to the end of the scale, noting the angular displacement of the plumb line for each position. Repeat for the movement of the traversing mass to the left of the centre.8) Move the sliding mass higher up the mast and again measure the inclination for different positions if the traversing mass.

Calculations

L0.35M

b0.2M

B0.1M

p3070.307kg

w1.52kg

y1910.091M

di250.025M

Y20.103m

di250M

I0.000233333

V0.00175

BM0.133333333

BG0.0785

bg20.0905

Theoretical calculations LOW MASS1) , = 0.0785M

2) = = =

3) = = =

4) = =

5) ==

Calculations

Experimental calculations Low Mass: angles on the left side denoted with a symbol and multiplied by -1 to negate the minus

Right side (+):X= 0.02M,X= 0.04M,X= 0.06M,X= 0.08M,Left side (-):X= 0.02M,X= 0.04M,X= 0.06M,X= 0.08M,

Calculations

Data recorded.

L0.35M

b0.2M

B0.1M

p3070.307kg

w1.52kg

y1910.091M

di250.025M

Y20.103m

di250M

I0.000233333

V0.00175

BM0.133333333

BG0.0785

bg20.0905

Theoretical calculations MID mass1) , = 0.0905M

2) = = =

3) = = =

4) = =

5) ==

Experimental calculations MID mass angles on the left side denoted with a symbol and multiplied by -1 to negate the minus Right side (+):X= 0.02M,X= 0.04M,X= 0.06M,X= 0.08M,Left side (-):X= 0.02M,X= 0.04M,X= 0.06M,X= 0.08M,

Percentage Differences LOW MASS height = MID MASS height =

Results

Discussion

Looking at the experimental metacentric height vs distance graph, the trendline is very linear. The GM is higher on the left hand side (-) compared to the right hand side because there was more weight on the left side in a nylon screw.The theoretical GM values are positive for the low-mass and the Mid-mass. This means that the pontoon is stable and would not shake violently and roll over. It also means our rig would roll slowly and return to the original position slowly compared to if the GM values were a minus/ zero figure. Looking at the angles deflection, the further out our traversing mass was, the more our rig would roll. In the results, at Low-mass, with the traversing mass out at 0.02M, a 2.5 degree roll was achieved. When the sliding mass was raised to 0.237M mid-mass, and traversing mass at 0.02M out, an angle of 3.4 degrees was got.

Conclusion

When our rig was in water, it did not roll over and the experimental data recorded had positive GM values for the sliding mass at the two different points on the plumb line. From the theory calculations and graph plotted, both the GM values for mass at low-point and mid-point were positive figures, meaning in theory the pontoon would be stable enough to not roll over when submerged in a body of liquid as expected.Our angle deflection was more severe when the sliding mass was placed at the middle of the plumb line (y=0.103M) compared to bottom (y=0.093M). However there was greater angles on the left side of the pontoon compared to the right when moving the traversing mass out towards the left or right of the rig. This was due to more mass on the left side as there was a nut and bolt on the left side. This also changed the centre of gravity of the rig.If I was to redo this experiment, I would sort out the imbalance with the screw and nut making the rig weight biased to the left side.GM is only an indicator for stability. It should not be relied upon whole heartily in the real world as to have the highest stability when designing boats, ships etc. Designers should conduct more tests that forces the floating mass into severe angles that would be subjected to in ocean storms.

Supplemented questions

1. Check weather the results you got conform to Archimedes Principle. Archimedes principle says that if the weight of the object is heavier than the weight of the water it will sink, and that the mass of the water dispersed when the object is immersed is the same as the mass of the object itself. the results I got proves this as the rig floated an did not sink. Thus proving the rigs mass was not greater than the mass of the water. 2. If the weight is loaded higher on the barge, does this change the Metacentric radius (BM)? The BM does not change, but the BG does as the formula for . This then affects the GM, as GM = BM BG.BM relies on the shape of our pontoon as it is a restoring moment. The wider our pontoon, the more stability it has.

3. In which direction is the barge more stable (roll or pitch)? Calculate the theoretical Metacentric height in the fore-aft pitch direction.

The barge is more stable in pitch as the length is longer than the width, thus giving more stability (ships roll over to their sides, not to their front or backs because the greater length front to back provides more stability).By switching the dimensions, the GM became smaller. This means that our pontoon will slowly roll, and return to it prior state slowly. The movement is not violent, having more stability.

Sources of reference http://www.navweaps.com/index_tech/tech-009.htm Dr Ken maloney & Colm Costelloe http://www.reptiles1.com/images/empty_tub.jpg) http://www.satwikscales.com/images/products/big/weighing_scale_IQ_100/weighing_scale_IQ_100.jpg) http://cdn.dickblick.com/items/554/30/55430-1012-1-3ww-l.jpg) http://accessscience.com/loadBinary.aspx?alD=3672&filename=330900FG0040.gif) http://en.wikipedia.org/wiki/Metacentric_height