HOW MANY BALLOONS DOES IT TAKE TO LIFT A HOUSE?Presented by Kim Nguyen

INTRODUCTION

Up video clip

http://s107.photobucket.com/albums/m303/kn_vi3t/?action=view&current=video-1.flv

Let’s pretend this is a perfect world where there is no wind or wind resistance, that the temperature remains the same at

different altitudes and that the acceleration of gravity is 9.81.

THEORY- ARCHIMEDES’ LAW: BUOYANCY FORCE

A body immersed in a fluid will experience a vertical upward force equal to the weight of the fluid it displaces.

The helium contained in the balloon is lighter than the air it displaces.

Solids experience a buoyancy force, but because the density of solids compared to gas is very large, it can be regarded as nothing.

DERIVATION OF EQUATIONSFb = Wfluid displaced

Fb = mfluidgFb = ρfluidVfluidg

Fb = Buoyancy force (kg·m/s2 = N)W = weight of fluid displaces (kg·m/s2 = N)m = mass of fluid (kg)g = acceleration of gravity on earth (m/s²)ρ = Density of fluid (Pa)V = volume of fluid (m³)

To determine the density of dry air at a specific temperatureρ = p / R•T

ρ = Density of fluid (kg/m³)p = pressure (Pa)R = individual gas constant (J/kg°K)T = temperature in Kelvin (˚K)

Fb = (p / R•T )(V)(g)

DERIVATION OF EQUATIONS

To calculated air pressure:p = 101325(1-2.5577E-5•h)^5.2588

p = pressure (Pa)

h = altitude above sea level (m)

To calculate the volume of an ellipsoid:

V = 4/3rlrwrh

Therefore the equation for the upward buoyancy force is:

Fb = (101325(1-2.5577E-5•h)^5.2588 / R•T)•(4/3rlrwrh)•(g)

DERIVATION OF EQUATIONS

To calculate the downwards force of the helium balloon:

FH = [(101325(1-2.5577E-5•h)^5.2588 / R•T)•(4/3rlrwrh)+mrubber+mstring]•(g)

Fg = mg

Newton’s Second Law – Gravitational Force:

Fg = gravitational force on earth. (kg·m/s2 =N)m = mass of the object. (kg)g = 9.81 = acceleration of gravity (m/s2)

Proof that the equation works will be done on the board

HYPOTHESIS

One helium balloon was able to hold 8.2 grams of lego (0.0082 kg).

The average weight of a house is 45 360 kg Therefore ,

45 460 ÷ 0.0082 kg= 5 531 707.317 So the amount of balloons needed to lift a house

is about 5 531 708? The higher the altitude, the lower the pressure.

Low pressure causes a smaller buoyancy force and that means more balloons.

The technicians from the Pixar estimated a total of 23.5 million balloons….

OBSERVATIONS Mass of an average house:

≈ 45 360 kg Mass of latex balloon (without helium)

≈ 2.9 g = 0.0029 kg Mass of fishing wire per 4.14 metres

≈ 3.0 x10-5 kg Mass of fishing wire per 100 metres ≈ 7.246376812

x10-4 kg 7.2 x10-4 kg Horizontal circumference of a filled balloon

≈ 81.5 cm = 0.815 m Vertical circumference of a filled balloon

≈ 89 cm = 0.89 m Individual gas constant of air = 286.9 Individual gas constant of helium = 2 077 Temperature = 293 ºK

CALCULATIONS Width of radius of balloon = height of radius of

balloon.

c= 2r

r= c/2

r = 0.815/2

r = 0.1297112786 m

r ≈ 0.13 m Length of radius : ≈ 0.14 m Gravitational force on an average sized house = Weight

of the house.

Fg = W = mg

Fg = (45 360)(9.81)

Fg = 444981.6 N [down]

CALCULATIONS Sample calculation for:

- 20 ºC = 273 ºK- 0 m above sea level- house weight of 45 360 kg

Fb=(101325(12.5577E5•h)^5.2588/R•T)•(4/3rlrwrh) •(g) =(101325/286.9•293)•(4/3(0.141647899) • (0.129711279)²•(9.81) ≈ 0.1180433921 N [up] ≈ 0.118 N [up]

CALCULATIONS

FH=[(101325(12.5577E5•h)^5.2588/R•T)•(4/3rlrwrh)+mrubber+mstring]•(g)

=[(101325/ 2077•293)•(4/3(0.141647899) •

(0.129711279)²)+0.0029+0.00072]•(9.81)

≈ 0.05181776052 N [down]

≈ 0.0518 N [down]

Fnet = Fb - FH

= 0.1180433921 [up] -0.05181776052 [down]

= 0.06622563161 N [up]

CALCULATIONS

Fg ÷ Fnet

= 444981.6 ÷ 0.06622563161

= 6 719 174.875

= 6 719 175 balloons

Altitude above sea level (m) p (Pa) R (J/kg°K) T (°K) Altitude above sea level (m) F net (N) [up] W of House - 45 360 (N) # of balloons needed

0 101325.00000000 286.9 293 0 6.62256313E-02 444981.6 6.71917491E+06

153 99500.43647373 286.9 293 153 6.43936339E-02 444981.6 6.91033528E+06

305 97714.23322107 286.9 293 305 6.26001530E-02 444981.6 7.10831489E+06

458 95942.58934708 286.9 293 458 6.08212909E-02 444981.6 7.31621433E+06

610 94208.37027176 286.9 293 610 5.90800061E-02 444981.6 7.53184757E+06

763 92488.46373493 286.9 293 763 5.73530921E-02 444981.6 7.75863313E+06

915 90805.06342360 286.9 293 915 5.56628330E-02 444981.6 7.99423198E+06

1068 89135.73033762 286.9 293 1068 5.39866985E-02 444981.6 8.24243031E+06

l Radius (m) w Radius (m) h Radius (m) g (m/s^2) F Buoyancy (N) [up] Mass of Houses (kg) Weight of Houses (N) # of balloons needed

0.141647899 0.129711279 0.129711279 9.81 1.18043392E-0110 000 98100 1.48129958E+06

0.141647899 0.129711279 0.129711279 9.81 1.15917779E-0120 000 196200 2.96259917E+06

0.141647899 0.129711279 0.129711279 9.81 1.13836857E-0130 000 294300 4.44389875E+06

0.141647899 0.129711279 0.129711279 9.81 1.11772896E-0140 000 392400 5.92519833E+06

0.141647899 0.129711279 0.129711279 9.81 1.09752534E-0150 000 490500 7.40649792E+06

0.141647899 0.129711279 0.129711279 9.81 1.07748847E-0160 000 588600 8.88779750E+06

0.141647899 0.129711279 0.129711279 9.81 1.05787690E-0170 000 686700 1.03690971E+07

0.141647899 0.129711279 0.129711279 9.81 1.03842921E-0180 000 784800 1.18503967E+07

Altitude above sea level (m) p (Pa) R (J/kg°K) T (°K)

0 101325 2,077 293

153 99500.43647 2,077 293

305 97714.23322 2,077 293

458 95942.58935 2,077 293

610 94208.37027 2,077 293

763 92488.46373 2,077 293

915 90805.06342 2,077 293

1068 89135.73034 2,077 293

l Radius (m) w Radius (m) h Radius (m) g (m/s^2) F Helium (N) [down]

0.141647899 0.129711279 0.129711279 9.81 5.18177605E-02

0.141647899 0.129711279 0.129711279 9.81 5.15241456E-02

0.141647899 0.129711279 0.129711279 9.81 5.12367037E-02

0.141647899 0.129711279 0.129711279 9.81 5.09516048E-02

0.141647899 0.129711279 0.129711279 9.81 5.06725284E-02

0.141647899 0.129711279 0.129711279 9.81 5.03957553E-02

0.141647899 0.129711279 0.129711279 9.81 5.01248568E-02

0.141647899 0.129711279 0.129711279 9.81 4.98562221E-02

GRAPHS

0 200 400 600 800 1000 12009.50000000E-02

1.00000000E-01

1.05000000E-01

1.10000000E-01

1.15000000E-01

1.20000000E-01

Buoyancy Force (N) [up]

F Buoyancy (N) [up]

GRAPHS

0 200 400 600 800 1000 12000.00000000E+00

1.00000000E+06

2.00000000E+06

3.00000000E+06

4.00000000E+06

5.00000000E+06

6.00000000E+06

7.00000000E+06

8.00000000E+06

9.00000000E+06

Number of balloons at different altitudes

# of balloons needed

GRAPHS

0

1000

00

2000

00

3000

00

4000

00

5000

00

6000

00

7000

00

8000

00

9000

000.00000000E+00

2.00000000E+06

4.00000000E+06

6.00000000E+06

8.00000000E+06

1.00000000E+07

1.20000000E+07

1.40000000E+07

Number of balloons for different houses

# of balloons needed

ERROR ANALYSIS

Because my experiment is based on a formula and calculations, it does not reflect on real life situations (who would use balloons to lift a house anyways?).

Possible miscalculations can occur during my calculations. Although I used Microsoft Excel to do the calculating to further reduce my errors.

SAFETY PRECAUTIONS

Let’s pretend that I was able to get 6 719 175 helium balloons and lifted my house, I should be careful: Of the weather, as the temperature does affect

the results. To attach the balloons to a secure area. ie. the

chimney. (that’s what Carl did) of birds, airplanes, blimps,…any aircraft. To not fall.

CONCLUSION

High altitude causes low pressure, which causes lower buoyancy force which makes it necessary for more balloons. The number of balloons needed to lift a

house varies between temperature, altitude, and weight of the house.

The number of balloons needed grow at a constant rate for both the difference in house weight and altitude.

REFERENCES Apple, Charles. How many balloons DOES it take to lift a house, anyway?. Visual Editors,

Dec. 12 2009 <http://www.visualeditors.com/apple/2009/05/how-many-balloons-does-it-take-to-lift-a-house-anyway/>. 

Goudie Christopher, Second year engineer student, Carleton University, conversation, Dec 11 2009.

Keisan. Volume of a ellipsoid. Casio Computer Co. LTD., Dec 12 2009 <http://keisan.casio.com/has10/SpecExec.cgi?path=05000000.Mathematics%2f0100020.Volume%20and%20surface%20area%2f13000600.Volume%20of%20a%20ellipsoid%2fdefault.xml&charset=utf-8> . 

Madrigal, Alexis. How Pixar’s Up House Could Really Fly. May 29 2009. Wired, 29 Sept. 2009 <http://www.wired.com/wiredscience/2009/05/how-pixars-up-house-could-really-fly/>.

  Nelson, David, ed. The Penguin Dictionary of Mathematics, Second Edition. England:

Penguin Books, 1998.

Shen Rastogi, Nina. How Many Balloons Would It Take To Lift A House?. Jun. 4 2009. Slate, Dec. 16 2009 < http://www.slate.com/id/2219775/>

The Engineering ToolBox. Homepage. Dec. 9 2008. The Engineering ToolBox, Dec. 13 2009 <http://www.engineeringtoolbox.com/>.

Taylor, Richard, Grade 12 physics teacher, Merivale High School, afterschool experiment, dec 10 2009.

White, Frank. Fluid Mechanics, Sixth Edition. New York: The McGraw-Hill, 2008.