Addison, Stephen R.

Gaiser, N.O (Jack)

Ross, J. G.

English

ScholarWorks@UARK

Journal of the Arkansas Academy of ScienceVolume 42 Article 361988Measurements of the Drag on Spheres FallingThrough the AirJ. G. RossUniversity of Central ArkansasStephen R. AddisonUniversity of Central ArkansasN.O ( Jack) GaiserUniversity of Central ArkansasFollow this and additional works at: http://scholarworks.uark.edu/jaasPart of the Fluid Dynamics CommonsThis article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able toread, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking priorpermission from the publisher or the author.This General Note is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the ArkansasAcademy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu, ccmiddle@uark.edu.Recommended CitationRoss, J. G.; Addison, Stephen R.; and Gaiser, N.O ( Jack) (1988) "Measurements of the Drag on Spheres Falling Through the Air,"Journal of the Arkansas Academy of Science: Vol. 42 , Article 36.Available at: http://scholarworks.uark.edu/jaas/vol42/iss1/36112 Proceedings Arkansas Academy off Science, Vol. 42, 1988Arkansas Academy off ScienceLITERATURE CITEDEINAM,G. E. 1956. Certain aspects of the natural history of themud turtle, Kinosternon baurii. Copeia 1956:186-188.IVERSON, J. B. 1979. The female reproductive cycle innorth FloridaKinosternon Baurii. Brimleyana 1:37-46.GIBBONS, J. W. 1970. Reproductive characteristics of a Floridapopulations ofmusk turtles (Sternotherus odoratus). Herpetologica26:268-270.MOLL,E. O., and J. M.LEGLER. 1971. The lifehistory of a neo-tropical slider turtle, Pseudemys scripta (Schopff) inPanama. Bull.Los. Ang. Co. Mus. Nat. Hist. Sci. 11:1-102.GROSS, D. 1982. Reproduction in a central Florida population ofSternotherus odoratus. M.S. thesis, University of South Florida,Tampa.TINKLE,D. W. 1961. Geographical variation in reproduction, size,sex, ratio, and maturity of Sternotherus odoratus. Ecology42:68-76.IVERSON, J. B. 1977. Reproduction in freshwater and terrestrial turtlesof north Florida. Herpetologica 33:205-212.WALTER E. MESHAKA, Department ofBiological Sciences, Arkansas State University, State University, Arkansas 72467.MEASUREMENTS OF THE DRAG ON SPHERES FALLINGTHROUGH THE AIRAproblem often investigated in undergraduate labs on classical mechanics is the effect of air resistance on falling objects. One of the mostinteresting of these involves the drag force on spheres falling through the air. Frequently in these experiments the time offallof the sphere is measured;this time is then used in a drag equation to calculate the distance fallen, for comparison with the measured distance. This distance measurementis a critical one and, for distances on the order of a few meters or less, must be accurate to within0.5 mm. Measurements to this precision aredifficult with the equipment and techniques available to the average undergraduate laboratory and, when several distances are involved, can presenta large source of error. The experiment to be described eliminates the need for this distance measurement by using a phototransistor timing circuitto obtain the velocities at various times of flight. These measured velocities are then compared to the velocities calculated from a drag equation.For many problems in fluidmechanics itis useful to define the dimensionless drag coefficient Cd in terms ofthe drag force Fd by the equa-tion Cd = (Fd/A)/(!/2()V2), where A is the cross sectional area of the body normal to the direction of motion, q is the density of the medium,and V is the speed of the body. Thus, for a sphere falling through the air, Fd = '/2Cd7rr2eaV2,where e a is the density of air and r is the radiusof the sphere. Another dimensionless quantity which relates the inertial forces on a body to the viscous forces is the Reynolds number R whichis defined by R = 2rV/»», where v is the kinematic viscosity of the medium. For spheres, the relationship between Cd and R is well known fromexperiment; a value of Cd m Vi for R between 103 and 10s is commonly accepted (H. Rouse, Elementary Mechanics ofFluids, Dover Publishing,1978, p. 249). For air at room temperature, v = 0.15 cm/s 2,and for a sphere of radius r = 2 cm, this range corresponds to velocities between38 cm/s and 3800 cm/s. Since all velocity measurements in this experiment are made in this range, the drag force may be approximated by:Fd = \v r2 a^ V2 [1]2 , [21*3 £s) . P];(1 - V^/Vo2) M[5]r O2 , [6][7]From Newton's 2nd Law,M 3F= Mg "Fd=Mg"T 7rand= g- (j*r25av2)/(|. ir t2qs ) ,with qs = the density of the sphere, so thatg|r = gd - (3gAV2)/(16gs gr)) = g l Wowhere Vo2 ¦ (16e sgr)/(3e.,). From this we find = gVo2- V2 Vo2Integrating both sides and setting V = 0 at t = 0, we have(l/V0 )tanh-l(V/V0 ) = gt/Voandtanh" ' (V/V0 ) = gt/V0With this approximation the velocity of a sphere falling through air is given byV = V0 {tanh(gt/V0 )} . [8]The device used for dropping the spheres was constructed by modifying the apparatus used ina typical undergraduate fallingbody lab. Anelectromagnet is mounted on a stand approximately 1.7 meters tall. Positioned in front of the magnet is a bracket designed to hold the ball beforeitdrops without influencing its motion when itis released. The ball is held against the bracket by a metal bar which is hinged to the stand undertension by a spring, and held in place by the electromagnet. Timing was done with a Merlan Micro Series Computimer #30M100 controlled bya Commodore CBMModel 8032 Computer. A double pole-single throw switch is used to control the electromagnet as well as the timer. When112Journal of the Arkansas Academy of Science, Vol. 42 [1988], Art. 36Published by Arkansas Academy of Science, 1988Proceedings Arkansas Academy off Science, Vol. 42, 1988 113General NotesTable 1.Velocities of falling spheres at various fall times.Time of fall Ideal Calculated MeasuredVelocity Velocity Velocity-l(msec) (cm/s) (cm/s) (cm/s)146 142.97 142.89 142.95 -2236 231.46 231.13 230.73278 272.21 271.67 271.98 _3324 317.17 316.33 316.50370 362.41 361.15 360.80-4406 398.14 396.47 396.23455 445.58 443.24 443.58-5484 473.72 470.91 470.57515 504.80 501.40 501.06F551 539.80 535.65 535.77igure 1. Differences from the ideal falling body case (dashed line -calculated values; plusses - measured values).l s; l l .the switch is thrown, the power to the magnet is cut off, allowing the bar to fall and release the ball. At the same time, the timer cycle begins.Located in the path of the ball is a photogate consisting of a laser beam directed onto a phototransistor connected to the timing system. Whenthe ball breaks the beam, the phototransistor is turned offand the timer stops. Inaddition, the computer can be set to measure the time the phototran-sistor is obstructed by the sphere. In this mode, a microsecond timer begins when the leading edge of the sphere breaks the beam and ends whenthe phototransistor is no longer darkened.The experiment was conducted using a plastic sphere ofdiameter d = 4.379 cm and mass M = 57.80 g. Since a photogate may not be obstructedby all of an object which passes through it, a preliminary test was made and the effective diameter of the sphere was found to be 4.27 cm. Themain timing procedure involved first setting the timer to the microsecond time-through-gate mode and determining the time taken for the ballto travel through the photogate. Great care was taken through numerous horizontal adjustments ofthe photogate system to insure that the spherepassed through the beam along its diameter, since this path willgive the longest possible time measurement for the pass through the gate. Whenthis time was achieved consistently, itwas recorded as the gate time and the timer was reset to the (millisecond) time-of-flight mode. The ball wasthen dropped to determine the time taken to fall from rest to the photogate. This entire procedure was followed for 10 different heights at approx-imately 15 cm increments.The average velocity of an object travelling a distance d in a time t can be written as Vav = d/t. Ifthe velocity increases linearly with time,then the average velocity is equal to the velocity at t/2. For the time interval during which the sphere passes through the gate, the accelerationis very nearly constant, so these approximations are valid for calculating the velocity at a particular time. Thus, the experimental velocity, whichis just the average velocity of the sphere through the photogate, may be calculated from 4.27 cm/(gate time) and corresponds to a total traveltime equal to the time of flight plus one half the gate time. Using this travel time, the theoretical velocity was calculated from the drag equation(Eq. 8). A value for g for the local latitude and elevation equal to 979.7 cm/s 2 and an air density of ga = 1.2 x 10" 3 g/cm 3 were used. A com-puter program was written to determine the lag time between the start of the timer and the actual dropping of the sphere to obtain the best agree-ment between the theoretical and calculated velocities; the lag was found to be 3 msec. The velocity ifthere were no air resistance is referred toas the ideal velocity and can be calculated from V = gt. Figure 1 shows the differences from the ideal case for the measured velocities and thevelocities calculated from Eq. 8 for various times of flight. Itis seen from the figure that the experimental velocities correspond very well withthe calculated velocities.We have shown that this timing technique can be used to provide measurements of the effects of drag forces on falling bodies to relativelyhigh precision. This can be useful in the undergraduate laboratory to supplement the usual free-fall measurement of the gravitational accelerationand to introduce the concept of drag forces in viscous media.J. G. ROSS, S. R. ADDISON, and N. O. GAISER, Physics Department, University of Central Arkansas, Conway, Arkansas 72032.EXTRALIMITALHUMMINGBIRDS INARKANSASThe Ruby-throated Hummingbird (Archilochus colubris) is the only species of hummingbird known to nest in Arkansas or anywhere elsein eastern North America (James and Neal, 1986; A.O.U., 1983). Until 1985 it was the only hummingbird species ever identified in Arkansas,where itis a common summer resident and migrant in all parts of the state (James and Neal, 1986).Before 1985 there were several well documented Arkansas reports (notably at LittleRock in December of 1978 and January of 1979 andat North Little Rock in October of1984) of unidentified hummingbirds that clearly belonged to other species (Arkansas Audubon Society files).Allthese were identified as members of the genus Selasphorus, most likely Rufous Hummingbirds (S. rufus), a western species that normally wintersinMexico, though small numbers of Rufous Hummingbirds regularly overwinter along the Gulf Coast of the U.S. (A.O.U., 1983). Positive iden-tification of the Arkansas birds could not be made on the basis offield observations or photographs. Only in the plumage of the adult male (withan all-rufous back) is S. rufus distinguishable in the field from the very similar Allen's Hummingbird (S. sasin). Infemale and immature plumages,113Journal of the Arkansas Academy of Science, Vol. 42 [1988], Art. 36http://scholarworks.uark.edu/jaas/vol42/iss1/36

Measurements of the Drag on Spheres Falling Through the Air

UARK (University of Arkansas )

Journal of the Arkansas Academy of Science Volume 42 Article 36 1988 Measurements of the Drag on Spheres Falling Through the Air J. G. Ross University of Central Arkansas Stephen R. Addison University of Central Arkansas N.O (Jack) Gaiser University of Central Arkansas Follow this and additional works at: https://scholarworks.uark.edu/jaas  Part of the Fluid Dynamics Commons Recommended Citation Ross, J. G.; Addison, Stephen R.; and Gaiser, N.O (Jack) (1988) "Measurements of the Drag on Spheres Falling Through the Air," Journal of the Arkansas Academy of Science: Vol. 42 , Article 36. Available at: https://scholarworks.uark.edu/jaas/vol42/iss1/36 This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This General Note is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu. 112 Proceedings Arkansas Academy off Science, Vol. 42, 1988Arkansas Academy off ScienceLITERATURE CITEDEINAM,G. E. 1956. Certain aspects of the natural history of themud turtle, Kinosternon baurii. Copeia 1956:186-188.IVERSON, J. B. 1979. The female reproductive cycle innorth FloridaKinosternon Baurii. Brimleyana 1:37-46.GIBBONS, J. W. 1970. Reproductive characteristics of a Floridapopulations ofmusk turtles (Sternotherus odoratus). Herpetologica26:268-270.MOLL,E. O., and J. M.LEGLER. 1971. The lifehistory of a neo-tropical slider turtle, Pseudemys scripta (Schopff) inPanama. Bull.Los. Ang. Co. Mus. Nat. Hist. Sci. 11:1-102.GROSS, D. 1982. Reproduction in a central Florida population ofSternotherus odoratus. M.S. thesis, University of South Florida,Tampa.TINKLE,D. W. 1961. Geographical variation in reproduction, size,sex, ratio, and maturity of Sternotherus odoratus. Ecology42:68-76.IVERSON, J. B. 1977. Reproduction in freshwater and terrestrial turtlesof north Florida. Herpetologica 33:205-212.WALTER E. MESHAKA, Department ofBiological Sciences, Arkansas State University, State University, Arkansas 72467.MEASUREMENTS OF THE DRAG ON SPHERES FALLINGTHROUGH THE AIRAproblem often investigated in undergraduate labs on classical mechanics is the effect of air resistance on falling objects. One of the mostinteresting of these involves the drag force on spheres falling through the air. Frequently in these experiments the time offallof the sphere is measured;this time is then used in a drag equation to calculate the distance fallen, for comparison with the measured distance. This distance measurementis a critical one and, for distances on the order of a few meters or less, must be accurate to within0.5 mm. Measurements to this precision aredifficult with the equipment and techniques available to the average undergraduate laboratory and, when several distances are involved, can presenta large source of error. The experiment to be described eliminates the need for this distance measurement by using a phototransistor timing circuitto obtain the velocities at various times of flight. These measured velocities are then compared to the velocities calculated from a drag equation.For many problems in fluidmechanics itis useful to define the dimensionless drag coefficient Cd in terms ofthe drag force Fd by the equa-tion Cd = (Fd/A)/(!/2()V2), where A is the cross sectional area of the body normal to the direction of motion, q is the density of the medium,and V is the speed of the body. Thus, for a sphere falling through the air, Fd = '/2Cd7rr2eaV2,where e a is the density of air and r is the radiusof the sphere. Another dimensionless quantity which relates the inertial forces on a body to the viscous forces is the Reynolds number R whichis defined by R = 2rV/»», where v is the kinematic viscosity of the medium. For spheres, the relationship between Cd and R is well known fromexperiment; a value of Cd m Vi for R between 103 and 10s is commonly accepted (H. Rouse, Elementary Mechanics ofFluids, Dover Publishing,1978, p. 249). For air at room temperature, v = 0.15 cm/s 2,and for a sphere of radius r = 2 cm, this range corresponds to velocities between38 cm/s and 3800 cm/s. Since all velocity measurements in this experiment are made in this range, the drag force may be approximated by:Fd = \v r2 â V2 [1]2 , [21*3 £s) . P];(1-V^/Vo2) M[5]r O2 , [6][7]From Newton's 2nd Law,M 3F= Mg "Fd=Mg "T 7rand= g - (j*r25av2)/(|. ir t2qs ) ,with qs = the density of the sphere, so thatg|r = gd - (3gAV2)/(16gs gr)) = g l Wowhere Vo2 ¦ (16e sgr)/(3e.,). From this we find = gVo2-V2 Vo2Integrating both sides and setting V = 0 at t = 0, we have(l/V0 )tanh-l(V/V0 )= gt/Voandtanh"'(V/V0 )= gt/V0With this approximation the velocity of a sphere falling through air is given byV = V0 {tanh(gt/V0 )} . [8]The device used for dropping the spheres was constructed by modifying the apparatus used ina typical undergraduate fallingbody lab. Anelectromagnet is mounted on a stand approximately 1.7 meters tall. Positioned in front of the magnet is a bracket designed to hold the ball beforeitdrops without influencing its motion when itis released. The ball is held against the bracket by a metal bar which is hinged to the stand undertension by a spring, and held in place by the electromagnet. Timing was done with a Merlan Micro Series Computimer #30M100 controlled bya Commodore CBMModel 8032 Computer. A double pole-single throw switch is used to control the electromagnet as well as the timer. When112Journal of the Arkansas Academy of Science, Vol. 42 [1988], Art. 36Published by Arkansas Academy of Science, 1988Proceedings Arkansas Academy off Science, Vol. 42, 1988 113General NotesTable 1.Velocities of falling spheres at various fall times.Time of fall Ideal Calculated MeasuredVelocity Velocity Velocity -l(msec) (cm/s) (cm/s) (cm/s)146 142.97 142.89 142.95 -2236 231.46 231.13 230.73278 272.21 271.67 271.98 _3324 317.17 316.33 316.50370 362.41 361.15 360.80-4406 398.14 396.47 396.23455 445.58 443.24 443.58-5484 473.72 470.91 470.57515 504.80 501.40 501.06F551 539.80 535.65 535.77igure 1. Differences from the ideal falling body case (dashed line-calculated values; plusses- measured values).l s; l l .the switch is thrown, the power to the magnet is cut off, allowing the bar to fall and release the ball. At the same time, the timer cycle begins.Located in the path of the ball is a photogate consisting of a laser beam directed onto a phototransistor connected to the timing system. Whenthe ball breaks the beam, the phototransistor is turned offand the timer stops. Inaddition, the computer can be set to measure the time the phototran-sistor is obstructed by the sphere. In this mode, a microsecond timer begins when the leading edge of the sphere breaks the beam and ends whenthe phototransistor is no longer darkened.The experiment was conducted using a plastic sphere ofdiameter d = 4.379 cm and mass M = 57.80 g. Since a photogate may not be obstructedby all of an object which passes through it, a preliminary test was made and the effective diameter of the sphere was found to be 4.27 cm. Themain timing procedure involved first setting the timer to the microsecond time-through-gate mode and determining the time taken for the ballto travel through the photogate. Great care was taken through numerous horizontal adjustments ofthe photogate system to insure that the spherepassed through the beam along its diameter, since this path willgive the longest possible time measurement for the pass through the gate. Whenthis time was achieved consistently, itwas recorded as the gate time and the timer was reset to the (millisecond) time-of-flight mode. The ball wasthen dropped to determine the time taken to fall from rest to the photogate. This entire procedure was followed for 10 different heights at approx-imately 15 cm increments.The average velocity of an object travelling a distance d in a time t can be written as Vav= d/t. Ifthe velocity increases linearly with time,then the average velocity is equal to the velocity at t/2. For the time interval during which the sphere passes through the gate, the accelerationis very nearly constant, so these approximations are valid for calculating the velocity at a particular time. Thus, the experimental velocity, whichis just the average velocity of the sphere through the photogate, may be calculated from 4.27 cm/(gate time) and corresponds to a total traveltime equal to the time of flight plus one half the gate time. Using this travel time, the theoretical velocity was calculated from the drag equation(Eq. 8). A value for g for the local latitude and elevation equal to 979.7 cm/s 2 and an air density of ga= 1.2 x 10" 3 g/cm 3 were used. A com-puter program was written to determine the lag time between the start of the timer and the actual dropping of the sphere to obtain the best agree-ment between the theoretical and calculated velocities; the lag was found to be 3 msec. The velocity ifthere were no air resistance is referred toas the ideal velocity and can be calculated from V = gt. Figure 1 shows the differences from the ideal case for the measured velocities and thevelocities calculated from Eq. 8 for various times of flight. Itis seen from the figure that the experimental velocities correspond very well withthe calculated velocities.We have shown that this timing technique can be used to provide measurements of the effects of drag forces on falling bodies to relativelyhigh precision. This can be useful in the undergraduate laboratory to supplement the usual free-fall measurement of the gravitational accelerationand to introduce the concept of drag forces in viscous media.J. G. ROSS, S. R. ADDISON, and N. O. GAISER, Physics Department, University of Central Arkansas, Conway, Arkansas 72032.EXTRALIMITALHUMMINGBIRDS INARKANSASThe Ruby-throated Hummingbird (Archilochus colubris) is the only species of hummingbird known to nest in Arkansas or anywhere elsein eastern North America (James and Neal, 1986; A.O.U., 1983). Until 1985 it was the only hummingbird species ever identified in Arkansas,where itis a common summer resident and migrant in all parts of the state (James and Neal, 1986).Before 1985 there were several well documented Arkansas reports (notably at LittleRock in December of 1978 and January of 1979 andat North Little Rock in October of1984) of unidentified hummingbirds that clearly belonged to other species (Arkansas Audubon Society files).Allthese were identified as members of the genus Selasphorus, most likely Rufous Hummingbirds (S. rufus), a western species that normally wintersinMexico, though small numbers of Rufous Hummingbirds regularly overwinter along the Gulf Coast of the U.S. (A.O.U., 1983). Positive iden-tification of the Arkansas birds could not be made on the basis offield observations or photographs. Only in the plumage of the adult male (withan all-rufous back) is S. rufus distinguishable in the field from the very similar Allen's Hummingbird (S. sasin). Infemale and immature plumages,113Journal of the Arkansas Academy of Science, Vol. 42 [1988], Art. 36https://scholarworks.uark.edu/jaas/vol42/iss1/36

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Measurements of the Drag on Spheres Falling Through the Air

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