Modeling the viscous torque acting on a rotating object

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1 Modeling the viscous toque acting on a otating object Manon E. Gugel Physics Depatment, The College of Wooste, Wooste, Ohio 6 (Apil 0, ) By dawing an analogy between linea and otational dynamics, an equation descibing the viscous toque acting on an object otating in a fluid can be anticipated. Stokes and Newton s models of a viscous dag foce ae commonly used to descibe the damping foce acting on an object moving linealy though a fluid. This expeiment demonstates that these models can be extended to descibe the viscous toque that damps the otation of an object in a fluid. When the otating object is ound, the fluid flow is lamina; hence, the viscous toque is popotional to the angula velocity to the fist powe, analogous to Stokes model. Howeve, when the otating object is ough, causing the fluid flow to be tubulent, the viscous toque is popotional to the angula velocity to the second powe, analogous to Newton s model. In addition, this expeiment demonstates that the popotionality constant between viscous toque and angula velocity is dependent on the shape of the object, as is the case in both Stokes and Newton s models. INTRODUCTION As an object moves though a fluid, the viscosity of the fluid acts on the moving object with a foce that esists the motion of the object. Two common appoaches fo modeling this esistive foce ae the Stokes and Newton s models. In 5, Si Geoge Gabiel Stokes published equations of viscous flow. In paticula, Stokes detemined the esistive foce, o viscous dag foce, of a sphee falling unde the foce of gavity in a fluid, eithe liquid o gas, to be diectly popotional to the sphee s velocity. Stokes model most accuately descibes objects moving linealy in a fluid that moves with lamina o steady flow. Si Isaac Newton, howeve, showed that the dag foce is popotional to the squae of the velocity of the object and acts in the diections opposite to the diection of the velocity. Newton s model is associated with highe velocities and tubulent, o non-steady, flow. This expeiment investigates the elationship between fictional toque and angula velocity, by dawing an analogy between the linea quantities discussed above and measuable angula quantities. THEORY Most commonly, viscous dag is modeled by eithe Stokes o Newton s models. Both models agee that the dag foce acts in the diection opposite to the velocity vecto fo the object in motion. The viscous dag foce accoding to Stokes model fo an object in lamina flow is given by: = c v (), whee c is a popotionality constant that depends upon the viscosity of the fluid and the shape of the object and v is the instantaneous velocity of the object. The viscous dag foce accoding to Newton s model fo an object in tubulent flow is given by: = c v ˆ v (), whee c is a popotionality constant that again depends upon the viscosity of the fluid and the shape of the object. Based on these models, this expeiment allows fo the esistive dag foce to be popotional to some othe powe of v. Such a dag foce would be given by: = c v n v ˆ (), whee c is anothe popotionality constant depending on the fluid s viscosity and the object s shape, and n is the powe of the velocity.

2 By dawing an analogy between linea and otational dynamics, the otational countepat to such a dag foce can be anticipated. In paticula, by substituting the coesponding otational quantities into equation (), the fictional toque acting on an object otating with angula velocity ω might be modeled by: τ D = Iα = I dω dt = c ω n (). Then solving equation () fo dω/dt, yields: dω dt = c I ω n = kω n (5), whee k = c /I is a new popotionality constant. Diffeential equation (5) can easily be solved fo vaious values of the exponent n. Fo instance, if n =, equation (5) can be solved using the sepaation of vaiables technique to yield: ω = ω o e kt (6), whee ω o is the initial angula velocity. And, if n =, the solution to equation (5) is given by: ω = kt + (7). ω o EXPERIMENT This expeiment uses an Ealing ai gyoscope (#-0). The steel oto ball of the gyoscope is the otating mass that is studied. The angula velocity, o fequency of otation, of the oto ball is detemined using a lase. Black stips of electical tape ae placed on the uppe potion of the shiny oto, and a lase is aligned so that it eflects off this potion of the oto. Then the lase beam is focused though a lens onto the tiny pinpoint head of a fast photodiode. Figue () below illustates the cicuity setup fo the photodiode. powe supply (5 V) phototansisto Schmidt tigge S esisto fequency counte GPIB Gugel: Modeling viscous toque voltage eading to pass though the Schmidt tigge. But, when no light is incident on the phototansisto, little cuent makes it to the esisto, sending a lowe voltage eading to the Schmidt tigge. The Schmidt tigge then shapes the voltage signals into a shap squae wave, which can then be ead by a Hewlet Packad fequency counte (55A). In addition, the fequency counte is connected to a Macintosh compute with a GPIB cable, in ode fo a LabView algoithm to impot the data acquied though the fequency counte. Finally, a tank of compessed nitogen gas is connected by a tube to the ai gyoscope appaatus. The pessue of the N gas coming out of the tank is adjusted to a constant psi thoughout the expeiment, so that the oto sits on a cushion of gas. A LabView (v..) pogam Viscous Toque.LV, was witten to set the contols fo and to impot data fom the fequency counte. The oto is spun by hand to attain the highest possible initial angula velocity. Then, as the oto spins, the LabView pogam calculates the aveage ω value ove 0.0 second intevals and ecods a table of values fo ω and the coesponding time to a file designated by the use. In ode to moe thooughly investigate the elationship between fictional toque and angula velocity, the oto is alteed by adding suface aea to the od, which causes the otation to be damped moe quickly. Rectangula pieces of styofoam boad, each measuing 0.5 cm thick by.0 cm wide by 6.0 cm long, ae used fo the additional aea. In ode to help keep the od upight as it otates, two ectangula pieces ae added opposite one anothe (like wings spanning appoximately cm) to the od of the oto fo each un. Each pai of styofoam pieces is efeed to as A, so that the un with no additional aea is un 0A, with two additional pieces is un A, with fou additional pieces is un A, and with six additional pieces is un A. ANALYSIS AND INTERPRETATION compute with LabView Figue (). Cicuity involved with the photodiode. In ode to detemine if any of the data can be modeled by equation (6), a semi-log plot of ω vesus time fo all data is pefomed. When the lase light eflects onto the photodiode (o phototansisto), the cuent supplied by the powe supply easily passes though the phototansisto, causing a high

3 Gugel: Modeling viscous toque ω (angula velocity (ps)) 5 7 A data 00 A data 00 Time (seconds) linea fit ove egion ω ps to max ps A data 00 0A data Figue (). Semi-log plot of ω vesus time fo all data sets, whee a linea esult implies that the exponent n = in equation (). Since only the 0A data in figue () appeas to have a staight, linea esult on the semi-log plot, only the 0A data, o data taken without additional aea, is accuately modeled by equation () with n =. In ode to detemine the value of n fo the othe sets of data, equation (5) is consideed. Thus, dω/dt vesus ω is plotted, using Igo Po (v..0). A powelaw fit is pefomed to detemine the values of k and n fo each data set. In paticula, we ae inteested in fitting a function of the fom: y = c ( x) c (), The constant c epesents k, and c epesents n fom equation (5). Thus, on a log-log plot, the slope of a linea fit gives the exponent n and the intecept elates to the popotionality constant k, such that log(k) is the intecept. Table () below lists the outcomes of the powelaw fit, and figue () shows the log-log plot of dω/dt vesus ω fo all fou sets of data. All of the data is fit ove the moe accuate egion of highe ω values, since at smalle ω values all souces of eo have a geate impact. Specifically, the data is fit fom the maximum ω, o initial ω o, to appoximately ω = ps. Table (). k and n values (fom equation (5)) fo all fou data sets, including standad deviation based on the powelaw fit. data slope = n intecept = k 0A.0±0.0 (.±0.5)x0 - A.±0.0 (7.0±0.)x0 - A.0±0.05 (.0±0.)x0 - A.0±0.0 (7.±.)x0 - dω/dt (otations/second ) aea = A aea = A aea = A aea = 0A powelaw fit fo data with wings powelaw fit fo data with no aea 7 5 ω (angula velocity (ps)) Figue (). Log-log plot of dω/dt vesus ω fo all data, whee the slope gives the exponent n and the intecept gives the popotionality constant k fom equation (5). As shown in figue (), the data fo 0A has a diffeent slope, o n value, than does the data fo A, A, and A. Indeed, in table (), we see that n is appoximately fo the data with no additional aea, as expected. Also, as shown in table (), n is appoximately fo all of the data taken fo the oto with wings, o additional aea. Thus, the viscous toque is diectly popotional to angula velocity fo the oto and smooth od, i.e. τ D ω fo the 0A data, and the viscous toque is popotional to the angula velocity squaed fo the oto with wings, i.e. τ D ω fo the A, A, and A data. Since the A, A, and A data ae all modeled by the same equation, analysis of this data leads to an undestanding of any dependence in the popotionality constant k on the known physical system. Thus, consideing equation (7),

4 the popotional elationship between the toque and the angula velocity squaed can be demonstated. Consequently, a diect plot of the invese angula velocity vesus time will have a linea esult fo the data with wings, given τ D ω fo the A, A, and A data. Figue () below illustates this elationship. As in figue (), a linea fit is shown only fo the egion fom initial ω o to ω ps, whee the data is moe accuate. (ps) - ) - / ω..0 (angula velocity (7.7±0.)x (.±0.07)x Time (seconds) aea = A aea = A aea = A linea fit ove egion ω ps to max ps 000 Gugel: Modeling viscous toque (6.±0.0)x0-00 Figue (). Plot of /ω (angula velocity - ) vesus time fo data with additional aea, demonstating the popotional elationship between τ D and ω fo this data. The slopes of the linea fits in figue () give the popotionality constants k fom equation (7). Thus, we note that these slopes as well as the k values listed in table () ae consistent. In addition, the popotionality constant k appeas to incease as the amount of aea added to the od is inceased, indicating that the popotionality constant is diectly elated to the shape of the otating object. should be expected, since the ai flow aound the od as the oto otates will be steady and cicula. On the othe hand, when wings ae attached to the od, the viscous toque is best modeled by an equation analogous to Newton s model of viscous foce fo objects moving in tubulent flow. Tubulent flow is geneally associated with votices foming behind the object as it moves. And, indeed, it is plausible that votices ae ceated behind the wings attached to the od, as the oto spins. In addition, figue () eveals the fact that the popotionality constant between the viscous toque and the squae of the angula velocity depends upon the amount of suface aea added to the od. This is also expected, since in both Stokes and Newton s models the popotionality constant depends upon the shape of the object and the viscosity of the fluid. In this expeiment, the viscosity of the fluid(s), a combination of N gas and ai, emains constant, except fo vaiations due to tempeatue and pessue; thus, vaiations in the popotionality constant ae expected to be elated only to the shape, o geomety, of the object. And, as additional wings ae added to the od, the shape of the otating object is effectively changed. Robet E. Steet, Golie Encyclopedia (Golie Electonic Publishing, Inc., 5). Dwight E. Gay, Ph.D. (coodinating edito), Ameican Institute of Physics Handbook (McGaw-Hill Book Company, Inc., The Maple Pess Company, Yok, PA, 57); pp. -. Joseph Nowood, J., Intemediate Classical Mechanics (Pentice-Hall, Inc., Englewood Cliffs, NJ, 7); pp CONCLUSION Data analysis has shown that the shape of a otating object has an inteesting impact on the viscous toque. Fo the smooth od and oto, the viscous toque is best modeled by an equation analogous to Stokes model of viscous foce fo objects moving in lamina flow. This esult

5 J. Chem. Phys. Vol. 0, No. 0, May 6