FLUID MECHANICS PRACTICAL REPORT - Module 2.docx

FLUID MECHANICS PRACTICAL REPORT

MODULE H. 02

HYDROSTATIC PRESSURE

GROUP PI

Adam Yuta Prayoga 1206292370

Asti Diar Syafitri 1206292414

Bimasena Heribowo 1206292351

Christopher Kevinly 1206223846

Nathan 1206292420

Wednesson Lawijaya 1206230593

Date of Practicum : 28th of November 2013

Laboratory Assistant : Rahmat Fitrah

Date Approved :

Grade :

Assistant’s Signature :

Laboratory of Hydraulics, Hydrology and River

Department of Civil Engineering

Faculty of Engineering

University of Indonesia

Depok


1. Objective

To measure the magnitude of hydrostatic force on a vertical plane

To determine the relationship between water height and additional

mass of the apparatus

2. Theorem

Every object which is soaked in water will be objected with

perpendicular forces on its surface as much as ρ.g.h (ρ is the density of

water).

The magnitude of the hydrostatic force on a flat plane is:

F=ρ × g × A × ycg ……………… .. (1 )

And its working point from the water surface is:

Zcf=(Y cg+I cg

A × ycg)Sinθ …………. (2 )

Where:

ρ = Density of water

g = Gravitational acceleration

ycg = Distance of the COM from the water surface

A = Area of the flat plane

Icg = Moment of inertia of the flat plane in respect to the

horizontal axis which passing through the COM.

1 | H y d r o s t a t i c P r e s s u r e

m.g

b

yd

ar’

r

L

m.g

b

yd

ar’

r

L


ɵ = Angle of tilt of the plane in respect to the water surface

Zcf = The distance between the surface and the point of force.

For “partially submerged” condition, applies:

m × L=0.5 × ρ× b × y2(a+d− y3 )………. (3 )

m

y2=−ρb

6 L+

ρb ( a+d )2L

… …………. ( 4 )

For “totally submerged” condition, applies:

m × L=ρ × b ×d × ycg (a+ d2− d2

12× ycg)………. (5 )



ycg= y−d2

… …………… ..(6)

m= ρbhL [a+ d

2 ] y−ρb d2 (d+3a )

6 L……………. (7 )

3. Apparatus

Hydraulics bench

Hydrostatic pressure practical apparatus

Loading

Ruler

Calipers

Figure notations:

1. Vessel / tank

2. Nivo

3. Loading arm

4. Scale arm

5. Quadrant object

6. Arm installation screw

7. Scale arm

8. Sharp axis

9. Calibrating load



10. Water level scale

11. Rectangular plane surface

12. Draining tab

13. Supporting footings

4. Procedure

Measure the length a, L, d and b of the apparatus

Adjust the supporting footings so the vessel is perfectly level

Insert the loading plate on the tip of the scale arm

Adjust the calibrating load so that the scale is perfectly level

Add loadings on the loading plate

Close the draining tab and fill the vessel with water little by little

until the scale arm is level

Note the height of the water (y) on the data column

Redo step 5 to 7 until the weight is 370 grams

Unload the load with the same rate with the loading.

Reduce the water level so that the scale arm is perfectly level

Measure the height of water (y) on the data column

Redo steps 9 to 11 until the loading is completely removed

5. Assignment

Proof equation (3) and (5) by equation (1) and (2)!

Ans:

Equation 3:



ΣMo=0W . L=FHydrostatic . Zcf


m . g . L=ρ . g . A . ycg (a+d−y3 )

m . L= ρ . y .b . 0,5 y (a+d− y3 )

m . L=0,5. ρ . b . y2 (a+d− y3 )

Equation 5:


m . g . L=ρ . g . A . ycg ( ycg+I cg

A . ycg)

m . L=ρ . b .d . ycg (a+d2

+

112

. b . d3

b . d . ycg)

m . L=ρ . b .d . ycg (a+d2

+d2

12. ycg)

6. Data Processing

Experimental Data

Filling Tank Draining Tank Average

Mass (m)

Water Height

(y) (cm)

Mass (m)

Water Height

(y) (cm)

Mass (m)

Water Height

(y) (cm)(gram) (gram) (gram)

50 4.7 50 4.7 50 4.7



70 5.6 70 5.5 70 5.5590 6.3 90 6.3 90 6.3

110 6.9 110 6.9 110 6.9130 7.6 130 7.6 130 7.6150 8.2 150 8.2 150 8.2170 8.8 170 8.8 170 8.8190 9.3 190 9.3 190 9.3210 9.8 210 9.8 210 9.8230 10.4 230 10.5 230 10.45250 10.9 250 10.9 250 10.9270 11.3 270 11.4 270 11.35290 11.8 290 11.8 290 11.8310 12.4 310 12.5 310 12.45330 12.9 330 12.8 330 12.85350 13.4 350 13.3 350 13.35370 13.9 370 13.9 370 13.9

a = 10cm Green = Partially Submerged

b = 7.5 cm Blue = Totally Submerged

d = 10cm

L = 27.5 cm

Relationship between loading applied and height of water:

4 6 8 10 12 14 160

50100150200250300350400

Relationship Between Loading Applied and Height of Water

Height of water (cm)

Load

ing

Appl

ied

(gra

ms)



a) Partially Submerged:

Mass (m)

(grams)h (x) m/h2 (y) x2 y2 xy

50 4.7 2.263468 22.09 5.123286 10.638370 5.55 2.272543 30.8025 5.164451 12.6126190 6.3 2.267574 39.69 5.14189 14.28571

110 6.9 2.310439 47.61 5.338128 15.94203130 7.6 2.250693 57.76 5.065617 17.10526150 8.2 2.230815 67.24 4.976536 18.29268170 8.8 2.195248 77.44 4.819113 19.31818190 9.3 2.196786 86.49 4.825868 20.43011210 9.8 2.186589 96.04 4.781171 21.42857

∑ 67.15 20.17415 525.1625 45.23606

150.0535

a = 10 cm d = 10 cm

b = 7,5 cm L = 27,5 cm

4 5 6 7 8 9 10 112.122.142.162.18

2.22.222.242.262.28

2.32.32

f(x) = − 0.0193855022023031 x + 2.38620996894789

h vs m/h2 (Partially Submerged)


m/h

2 (g

r/cm

2)

y=bx+a

b=n(∑ xy )−(∑ x ) (∑ y )

n (∑ x2 )−(∑ x )2



b=−0.0194

a=(∑ y ) (∑ x2 )−(∑ x ) (∑ xy )

n (∑ x2 )−(∑ x )2

a=2.3862

y=−0.0194 x+2.3862

Theoretical b:

b theoretical=−ρb6 L

¿− 7.56 × 27.5

=−0.045

Theoretical a:

a theoretical=ρb(a+d )

2 L

¿ 7.5× 202× 27.5

=2.727

Relative Error:

relative error of b=|bPractical−btheoretical

b theoretical|×100 %

¿|−0.0194−(−0.045)−0.045 |×100%=56.89%

relative error of a=|aPractical−atheoretical

atheoretical|×100 %

¿|2.3862−2.7272.727 |×100 %=12.5 %



b) Totally Submerged:

DataWater Height (cm) x

Mass (m) y x^2 y^2 xy

1 10.45 230 109.2025 52900 2403.52 10.9 250 118.81 62500 27253 11.35 270 128.8225 72900 3064.54 11.8 290 139.24 84100 34225 12.45 310 155.0025 96100 3859.56 12.85 330 165.1225 108900 4240.57 13.35 350 178.2225 122500 4672.58 13.9 370 193.21 136900 5143∑ 97.05 2400 1187.633 736800 29530.5

a = 10 cm d = 10 cm

b = 7,5 cm L = 27,5 cm

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5200220240260280300320340360380

f(x) = 40.3606228940898 x − 189.624806483927

y vs m graph (Totally Submerged)


Load

ing

(gr)

y=bx+a



b=n(∑ xy )−(∑ x ) (∑ y )

n (∑ x2 )−(∑ x )2

b=40.361

a=(∑ y ) (∑ x2 )−(∑ x ) (∑ xy )

n (∑ x2 )−(∑ x )2

a=−189.62

y=40.361 x−189.62

Theoretical b:

b theoretical=ρbd (a+ d

2 )L

¿7.5× 10 × (10+5 )

27.5=40.91

Theoretical a:

a theoretical=ρb d2(3 a+d)

6 L

¿7.5× 102×(30+10)

6 × 27.5=181.82

Relative Error:

relative error of b=|bPractical−btheoretical

b theoretical|×100 %

¿|40.361−(40.91)40.91 |×100%=1.342%

relative error of a=|aPractical−atheoretical

atheoretical|×100 %



¿|189.62−181.82181.82 |× 100 %=4.29%

7. Analysis

Practical Analysis

In this experiment, the relationship which is wanted to be revealed

is the relationship between the loading implied and the height of water

which submerge the quadrant apparatus. In this experiment, the apparatus

which demonstrates the relationship between these two is in form of a

scale. This apparatus will measure the relationship of hydrostatic

pressure and weight implied by using moment stability. Since the water

will “push” the plane of the quadrant object and will cause a positive

moment, the weight which lies on the load will cause a negative moment.

The first thing which has to be done is to measure the dimension of

the demonstration apparatus. Note that the things that should be

measured are the length of the scale, the height of the quadrant object, the

width of the quadrant object, and the height from the plane to the scale

arm. These are needed in the calculation. The second part is ensuring that

the demonstration system is perfectly flat. Not leveled apparatus will

cause parallax error due to difficulties in reading the vessel’s scale. The

next step is to insert the loading plate on the end of the scale. Loading

plate will be used to contain loading. The next step is adjusting the scale

so that the scale is in equilibrium condition. This condition will cancel

out the other mass other than the loading and the hydrostatic force. The

next step is closing the draining tab which enables the vessel to be filled.

After closing the tab, add the loading (50gr) and fill the water into the

vessel. Note that the water should be added carefully, because if the

water touches the quadrant object directly, it will disturb the scale

reading. Measure the reading on the quadrant object immediately once



the scale reading is in the middle, this is to ensure that the equilibrium

condition has been reached when the reading is done. Repeat these steps

by adding 20 gr for each reading until the total loading reaches 370 gr. 20

gr is chosen in order to gain a fine gradation of data. After reaching 370

gr, continue by doing the draining measurement, by repeating the reading

for each subtraction of 20 gr. This is done in order to find the average of

both of them, so a more accurate set of data can be achieved.

Result Analysis

The data which has been obtained from the experiment is then

processed further. From the raw data, it is possible to know the

relationship between the height of water and the loading which has been

given. From the data, the height of water required increases as the

loading increases, while the detailed relationship will be explained in the

graph analysis.

From the data, it is possible to know the relationship between m/h2

and h can be determined. m/h2 is taken because in the equation (4) shows

that the m/h2 represents the y axis of the graph. In this relationship, it can

be known that both variables are linked linearly. The calculation which is

done is aimed to know the constants of the linear line of the graph. In

partially submerged condition, the b which is obtained is -0.0194 with an

a of 2.3862. The negative b indicates a negative slope. For the totally

submerged condition, the x axis represents the height of the water, while

the mass of the loading represents the y axis. The b is obtained from

equation (7), which reflects the rate of the change of hydrostatic force in

respect to depth. The b which is obtained is 40.361 and the a is 181.82.

Note that the b is positive, which indicates positive slope.

Graph Analysis

In this experiment, there are three graphs which are able to be

plotted. The first one represents the relationship between the height of

water and the weight applied, which represents the hydrostatic force



acting on the vertical plane. In this graph, each data of the experiment

shows a linear relationship. This shows that the experiment has been

done accurately, since the data is not scattered. This also shows that the

water level increase causes an increase of hydrostatic force acting on the

plane.

On the second graph, the relationship which is represented is the

relationship between mass divided by water height squared in the

partially submerged condition. In this graph, the line which is regressed

was directed downwards with negative slope. This is caused by water

height squared grows more quickly compared to mass of the load, so the

graph goes downward. In this graph, the data is a bit scattered, which

shows a relatively high error in the experiment.

On the third graph, the relationship which is represented is the

relationship between mass of the loading and the water height in the

totally submerged condition. Note that in this graph, the slope is positive,

which indicates the hydrostatic force (expressed as loading’s mass)

increases as the height of water increases. The data recorded was quite

linear, which indicates the experiment is done quite well.

Error Analysis

From the data which is obtained, the error can be determined. In

the first condition, which is partially submerged, the error recorded was a

bit high, with a relative error of a is 12.5% and the relative error of b is

56.89%. The second condition, which is fully submerged condition,

possess relatively low error, with a relative error of a is 4.29% and the

relative error of b is 1.342%.

The errors which occurred in the experiment may occur due to

some factors, such as stated below:

When pouring the water into the vessel, the quadrant object

may a little bit disturbed by the pouring

When pouring the water into the vessel, the water is a bit

excessive. In order to justify the measurement, the water



was drained out, so the data gained was not purely

classified as “filling tank”. The reverse also applies

It was a bit hard to read the scale perfectly, since the scale

sometimes oscillates in a high rate.

The water height reading may not totally accurate, since the

scales on the quadrant object is thin

The weight which was used was replaced several times

because the loading plate cannot accommodate very much

20 gr and 10 gr loads, so every 60 gr increase, the three 20

gr loads was replaced with a 50 gr and a 10 gr load.

8. Conclusion

This experiment is aimed to determine the relationship between

hydrostatic pressure and water level

There are two conditions which are observed, the first one is

partially submerged condition and the second one is the totally

submerged condition

The line equation which are regressed in the first condition is y=-

0.0194x+2.3862 and the second condition is y=40.361x-189.62.

As the water level increases, the hydrostatic pressure increases

9. References

Departemen Teknik Sipil Fakultas Teknik Universitas Indonesia.

2009. Modul Praktikum Mekanika Fluida dan Hidrolika. Depok :

Laboratorium Hidrolika, Hidrologi, dan Sungai

10. Attachment



Filling the water into the vessel


FLUID MECHANICS PRACTICAL REPORT - Module 2.docx

Documents

Transcript of FLUID MECHANICS PRACTICAL REPORT - Module 2.docx