Gallelleo Project

8/13/2019 Gallelleo Project

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School of Engineering and Materials Science

Identify the phenomenon that acts onfalling objects, affecting its fall time

Report on a problem by Galileo Galilei, to: Determine the phenomenon that affects the speed of a body that falls through that air. Perform appropriate experiments to measure the relevant factors affecting the drop time

for the body. Determine values of any relevant engineering coefficients. Develop a mathematical model that takes both gravity and the phenomenon into account

to accurately predict the fall time for different spherical objects.

Ram BHADRESA090155015

Dr. P. Wen


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dentify the phenomenon that acts on falling objects, affecting its fall time !"#$!#"!!%Ram &'(DR)*(

02/10/2009

School of Engineering and Ma erial!Science

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A"! rac

+he problem as to, determine the phenomenon that affects the speed of an abject thatfalls through that air, perform appropriate experiments to measure the relevant factorsaffecting the drop time for the object, determine values of any relevant engineering

coefficients and to develop a mathematical model that takes both gravity and the phenomenon into account to accurately predict the fall time for different sphericalobjects.

+he time taken for a ball of mass -." g and diameter -. cm, and a balloon of mass /. g anddiameter " cm, to fall from a height of .0 m as measured using a stop atch, and a tapemeasure. &oth ball and balloon as dropped seven times each in order to minimi1e experimentalerror. +he time for the ball to fall .0 m is consistently less than the time taken for the balloon tofall from the same height. +he average time taken for the ball and balloon to fall is $.! s and".%$s respectively. 2rom these values the terminal velocity of the ball and balloon ere found.+his can lead to a development of a mathematical model, hich can be applied to any sphericalobject at standard temperature and pressure ith the object having a coefficient of drag !.3/.


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#a"le of $on en !

(bstract.............................................................................................................................................i+able of 4ontents.............................................................................................................................ii$. ntroduction ..................................................................................................................................$

".+heory...........................................................................................................................................$5.(pparatus .....................................................................................................................................53.)xperimental procedure................................................................................................................5

.Ra results....................................................................................................................................3-.(nalysis of Data............................................................................................................................

-.$.6ist of 7ariables:....................................................................................................................-.".4alculated Results..................................................................................................................

-.".$.4alculation of aerodynamic drag ...................................................................................-.".".4alculation of acceleration to 78...................................................................................--.".5.4alculation of 78.........................................................................................................../-.".3.4alculation of time to 78...............................................................................................0

-.". .4alculation of time it takes bodies to travel .0 m......................................................$!-.5.9ncertainties ......................................................................................................................$3/.Discussion of Results..................................................................................................................$0.4onclusions.................................................................................................................................$/%.References...................................................................................................................................$/


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1. %n rod&c ion

+he aim of the experiment as to measure the time taken for a ball of mass -." g and diameter-. cm, and a balloon of mass /. g and diameter " cm, to fall from a height of .0 m using astop atch. +his ould lead to proof that a phenomena kno n as aerodynamic drag forces act

upon all bodies, regardless of si1e and that the si1e of the bodies vary the drag forces on them.+his ill also lead to a mathematical model being developed after analy1ing the data from theexperiment, to be able to accurately measure the drop time of different bodies.

e ill assume that there are only t o forces acting upon the body throughout the experiment.e ill also assume no other external forces act, for example there is no ind affecting the

bodies. (lso the viscosity and density of the fluid ill be a typical value based on the altitudeand temperature. 'ere the temperature is assumed to be "! 4 and altitude to be sea level. +hesmoothness of the bodies ill be assumed as e;ual.

2. #heor'

t is kno n that any surface#body in contact ith a flo ing fluid is subject to a force exerted bythe fluid. +his force is called a drag force.

+he body flo ing through the fluid, in this case air, is subject to t o forces, gravitationalattraction of the earth on the object, giving it eight


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+he amount of drag force that acts on a body ill depend on several factors. n this case thediameter of a perfectly shaped sphere. +he larger the body is, the greater surface area is incontact ith the fluid it is flo ing through, thus more collusions bet een the body andmolecules making up the fluid, hence a greater drag force the body ill suffer.

+he speed of the flo past the body is going to have an affect on the drag force. 2or example ahigh ind is more likely to blo a ball across a smooth table than a gentle bree1e. +his is

because the speed of air molecules hitting the ball is much greater> therefore hen the airmolecules impact upon the surface of the ball they provide a higher drag force. (lso hen ridinga bike, pedalling it gets harder the faster the bike is moving. +his is associated ith the

phenomenon of terminal velocity mentioned earlier in the report.

?ther factors such as viscosity of the fluid, the density of fluid and the smoothness of the bodyalso affect the drag force on bodies moving through a fluid. n this experiment, these factors

ere not necessary to measure and use in ma thematical calculations, as the differences theyould provide to the overall outcome ould be negligible. (s a result of this, smoothness as

ignores and density of fluid as taken to be of air at sea level and "! 4.+he drag force can be accurately calculated by using a calculated value kno n as the Reynoldsnumber. +he Reynolds number can be calculated by using the formulae,

+his calculated value could be looked up on a log graph, of Reynolds number against the

coefficient of drag.

2or this experiment ho ever a value for the drag coefficient ill be a standard for a typicalsmooth sphere, ith a Reynolds number bet een $! and $! . +his ill give a coefficient ofdrag at !.3/. +his value is based on dimension normal to flo .


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(. A))ara &!

$. *mall ball of light eight material". nflatable balloon5. *top@ atch, accurate to !. s

3. Aeasuring tape, accurate to !.!$m. eighing scales, accurate to !.! kg

*. E+)erimen al )roced&re

S e) 1, +he height of the $ st floor me11anine to the ground as measured. +his ould be thedrop height of the ball and balloon. +he tape measure used as made of non@rigid material,hence ill not remain perfectly vertical hen measuring.

S e) 2, +he eighing scales ere po ered up and the light eight ball as eighed. +he massas recorded on a sheet of paper. +he balloon as inflated ith normal air and as then sealed.

+he si1es of ball and balloon ere purposefully different as described earlier, relative si1e of

bodies affect aerodynamic drag upon them. +he balloon as also eighed.S e) (, &oth ball and balloon ere taken to the $ st floor and a reference point as set up so thatthe drop height as the same each time.

S e) *, +he ball as released first, and the stop atch as started the moment the ball asreleased. (s the ball hit the floor the stop atch as stopped. +his as repeated seven times inorder to minimise any experimental error. (s the ball as falling do n vertically, it becamedifficult to kno hen exactly the ball hit the ground for someone standing on the $ st floor. +hismay have led to some experimental timing inaccuracies. (s the ball also fell relatively fast,reaction times ill also be a source of error in time recording.

S e) 5, +he balloon as then released from the same height. (gain the stop atch as started

hen the balloon as released and stopped hen it hit the ground. +his as also repeated seventimes to minimise inaccuracies and errors ith timings. 'ere there as a problem ith indcurrents passing through the building. +he balloon ould drift to one side as a draught ble .+his as a major source of experimental error.

S e) -, +he balloon and ball ere again released from the same height for a second set ofrepeats. +his as to see hether the times recorded in the first instance ere accurate andconcise. + o sets of seven repeated results ere no recorded.


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5. Ra re!&l !

(ssumed temperature: "!B 4

(ssumed pressure: $.""-kg#m

BALLMass/kg Diameter/m0.00625 0.065 Set 1 Set 2

1.07 1.001.01 0.971.19 1.030.97 0.971.10 1.181.12 0.931.06 1.06

BALLOONMass/kg Diameter/m0.00750 0.250 Set 1 Set 2

2.89 2.992.94 1.62.94 2.253.03 2.732.88 2.972.99 2.382.75 2.49


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-. Anal'!i! of Da a

-.1. i! of aria"le!, D @ drag force in CC D @ drag coefficient

D @ diameter of sphere in metres @ density of air in kg#mV @ terminal velocity of body relative to fluid in m#sV @ velocity of body relative to fluid in m#st - +ime taken for bodies to drop .0 m, in secondsT @ atmospheric temperature in B4

-.2. $alc&la ed Re!&l !n order to analyse the ra data, a series of calculations ere made and thus a mathematical

model as produced.

-.2.1. $alc&la ion of aerod'namic drag



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-.2.2. $alc&la ion of accelera ion o V


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-.2.(. $alc&la ion of V


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-.2.*. $alc&la ion of ime o V


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-.2.5. $alc&la ion of ime i a e! "odie! o ra el 5.35m


$!


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$$


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$5


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-.(. 4ncer ain ie!

+he uncertainty associated ith each of the measured variables is given in the table belo . +hesevalues ere chosen based on the stated accuracy of the instrument, if available. ?ther ise, theyare reasonable estimates based on values typically reported.


4ncer ain ' De!cri) ion S'm"ol &merical al&e

*mall ball diameter b !.!! m

nflatable balloon diameter B !.!! m

+ime t !. s

Aass m !.! kg

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Determination of the Drag 4oefficient of a *phereoe 4ollege *tudent

Aarch "$, "!!")CGR 5 $ E )xperimental Aethods

6. Di!c&!!ion of Re!&l !

hen the drag force as calculated, the force exerted on the ball as significantly less than theforce applied onto the balloon. +his matches the predictions made by the background theory. +hesmaller ball has less contact ith the air as it falls. +here are less air molecules to be moved out

of the ay of the path of the ball as it travels. (s a result of this there is less impulse exerted onthe ball by the air molecules and vice versa. 2or the balloon ho ever, there are more moleculesneeding to be pushed out of the ay of its fall path. +he impulse ill be greater> therefore theopposing force exerted ill be more.

(s as predicted, the acceleration of the ball is greater than the ball. f the net force is taken intoaccount, this consists of the do n ard force, the eight and the opposing drag force. +he greaterthe drag force is the smaller the net or resultant force ill be. +his is sho n on page -, in acalculation to find the net thrust for the ball and balloon. 2rom this net force values, a simplemathematical relationship, bet een force, acceleration and mass is used to calculate theacceleration of both ball and balloon. +he acceleration for the ball is greater than the accelerationof the balloon because of the physics explained in the first paragraph.

(t terminal velocity, the eight force is exactly e;ual and opposite of the drag force.(cceleration is 1ero and velocity of the body is at itFs maximum. &y using the relationship, theterminal velocities for both ball and balloon ere found. +his resulted in t o different values ofterminal velocity being found. +he velocity of the ball as much higher than the velocity of the

balloon. +his as also due to the phenomenon of drag, as explained in paragraph one.

+he time to reach terminal velocity as also found. +his took into account a simple e;uation ofmotion, involving terminal velocity, acceleration and time. +he acceleration used as the onecalculated earlier. t as in this form, as the acceleration is not constant. +he acceleration of the

body ill start off as g , but ill then decrease as the force of drag comes into play, as thevelocity increases. e can gather from this that thee acceleration depends on the drag force and

the velocity of the body. &y using various mathematical relationships, the time for both ball and balloon to fall .0 m as found. +he time for the ball to fall as less than the balloon. +his as because of the reasons explained earlier in this paragraph and in paragraph one.

+he graphs on pages $$ and $" ere used to calculate the time taken for the bodies to fall .0 m.+o find the time for this, the area under the curve as found by a method of integration kno n astrape1ium rule. +his rule as then extrapolated through the graph and the area as progressivelyincreased, until the distance#area .0 m as found. (t this point the time taken as read off andrecorded. +his as repeated for the balloon. t as found that the time taken for the ball to fall

as less that the time taken for the balloon to fall through .0 m.

e can explain these results by using some of the scientific theory, mentioned earlier in the

report. (s the ball is released, it begins to accelerate at g . (fter a certain time period, the velocityincreases significantly. +he drag force than increased greatly as the drag force is proportional tothe velocity s;uared. (s the velocity continues to increase the drag force does that same. +heacceleration decreases also as this is taking place. +here then reaches a time hen theacceleration is 1ero, so the velocity cannot increase any further. (s the drag depends on velocity,this is also no stable. (t this point the do n ard force < eight= is e;ual to the opposing force


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During the experiment there ere some problems that ere encountered. (s the ball as fallingdo n vertically, it became difficult to kno hen exactly the ball hit the ground for someonestanding on the $ st floor. +his may have led to some experimental timing inaccuracies. (s the

ball also fell relatively fast, reaction times ill also be a source of error in time recording. +hetape measure used as made of non@rigid material, hence ill not remain perfectly vertical hen

measuring. +here as a problem ith ind currents passing through the building. +he balloonould drift to one side as a draught ble . +his as a major source of experimental error.

n future it ould be beneficial to use a building ith les ind draught, and using t ostop atches instead of one. 2or example there could be one person standing on the ground floor,timing the fall time and another person on the $ st floor doing the same. e then minimise the

problem ith kno ing hen the body hits the floor. 'o ever doing this brings otherinaccuracies, as the reaction time needed to start the stop atch from the ground floor, henindicated by the person on the $ st floor ill have to be very good, other ise there ill be a timedelay. ( rigid tape measure should have been used to measure the distance, as it ould be farmore accurate.

n future, there should be more si1es of ball and balloon to be used> a range of heights can also be used. +he mass of ball and balloon used can also be varied. +he formulae can then be re@tested and see if it still orks. +o make the data collected ore accurate, the density of the roomshould be measured using a barometer, and temperature should be measured using athermometer.

e have established that the time taken for a ball to fall form a given height of .0 m issignificantly shorter than that of a balloon. e have established the phenomenon that acts of the

bodies is aerodynamic drag, and the mathematical model is produced in the analysis of data. +heengineering coefficients are also found by using mathematical formulae and various publishedtexts. (ll the aims have been met and analysed.


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3. $oncl&!ion!

+he results of this experiment sho that the time taken for a small sphere, for example a ball isless than the time taken for a larger sphere to fall the same height, regardless of mass. +he

phenomenon that affects the fall time is kno n as aerodynamic drag.

9. Reference!

idden, A.

Gallelleo Project

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