TP B.2 Rolling resistance, spin resistance, and "ball turn"

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Harold King
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1 echnical proof TP B. olling reiance, pin reiance, and "ball urn" upporing: The Illuraed Principle of Pool and Billiard hp://billiard.coloae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") echnical proof originally poed: 1/14/008 la reiion: /1/008 Here are he force acing on a rolling ball: mg m: ball ma m 6oz, -a I: ball ma momen of ineria : ball radiu 1.1in r mg r : coefficien of rolling reiance d mg The ranlaional equaion of moion i: ΣF x = ma x μ r mg = m( a) ( 1) The ball acceleraion "a x " i negaie becaue he ball i deceleraing. So he coefficien of rolling reiance (CO) i relaed o he ball' deceleraion according o: a μ r = ( ) g The rae of deceleraion (a) can eaily be deermined experimenally by iming how long (T) i ake a rolling ball o rael a cerain diance (L) o a full op.

2 From baic kinemaic, he diance i relaed o deceleraion and ime wih: 1 L = ΔT a ΔT = ( aδt) ΔT 1 a ΔT 1 = a ΔT ( 3) o he deceleraion rae can be deermined wih: a = L ΔT ( 4) Subiuing Equaion 4 ino Equaion allow u o calculae he CO from experimenal meauremen: L μ r = gδt ( ) I ran a few experimen on a ypical cloh and go an aerage alue of: μ r 0.01 Thi number will obiouly ary wih cloh ype and condiion. A "fa" cloh will hae a lower CO, and a low cloh wih hae a higher CO. eurning o he diagram aboe, auming he ball radiu i much larger han he cloh dimple ize, he roaional equaion of moion (uing Equaion ) i: T = Iα dm g μ r mg m a = ( 6) d μ r mg = μ r mg So, from Equaion 6, he diance he erical force i hifed forward i: 7 d μ r d 0.016in ( 7)

3 The force in he diagram on he preiou page can be repreened by reulan normal force N and angenial fricion force F a hown below, where he normal force ac hrough he cener of he ball: T r z y x F F N From Equaion 6, he angenial fricion force, which creae orque T, i: F = T = μ r mg ( 8) Equaing he horizonal force componen from boh diagram gie: Nin( θ) Fco( θ) = μ r mg ( 9) Equaing he erical componen gie: Nco( θ) Fin( θ) = mg ( 10) Eliminaing N from Equaion 9 and 10 gie: F = mg in( θ) μ r co( θ) ( 11 ) Subiuing Equaion 8 gie: μ r co( θ) in( θ) = μ r ( 1) Thi i of he general form: Aco( θ) Bin( θ) = C ( 13) where A=- r, B=1, and C=/ r. Equaion 13 ha he oluion (from angen-half-angle ubiuion): θ 1 aan B B A C = and θ A C aan B B A C = ( 14) A C

4 Therefore, chooing he acue angle oluion, he angle of he reulan normal i: Simplifying gie: 1 1 μ r μ r θ = aan ( 1) μ r μ r 1 θ aan 1 3μ r μ r 1 θ 0.80deg ( 16) Thi i he angle a which he effecie cener of preure normal force ac. I will aume ha when he ball alo ha idepin, he reulan force relaed o rolling reiance (N and F) do no change ery much. When a ball pin in place, a fricion orque deelop in he cloh dimple which oppoe he moion. From pin-down e wih a ideo camera, ypical meaured deceleraion rae are approximaely: α mea 10 rad ec ( 17) Thi correpond o a fricion orque of: T = Iα mea T m α mea T inlbf ( 18) A hown in he diagram aboe, I will aume hi pin-down orque i aligned wih he cener-of-preure reulan normal ecor (N) while he ball i rolling. I will alo aume he orque magniude i approximaely he ame a wih he aic pin-down e. The hape of he ball-cloh dimple and he diribuion of fricion force wihin he dimple are no known; bu if he ball urn, he fricion force mu hae a ideway reulan ha make he ball urn (F in he diagram). If no, he analyi will predic a alue of 0 for F. A poible explanaion for hi force i ha he fricion force on he leading edge of he dimple migh be greaer han he fricion force on he railing edge. The force could alo repreen he cloh' reiance o any endency for mae pin o deelop a a reul of he orque and urning moion. If we aume he ball' elociy urn, and if we hae he z axi remain erical and he x axi remain in-line wih he ball' elociy, hen he angular elociy of he ball i: ˆj kˆ r ˆj kˆ ( 19) Here, I am auming no mae pin deelop while he ball i rolling. Een if here i a endency for mae pin o deelop, I am auming cloh force will deelop (a par of F ) o couner he pin endency.

5 The angular elociy of he frame i: where i he urn rae. k ˆ ( 0) If he ball' elociy urn, here mu alo be a urning force (F in he diagram) puhing he ball in he urn direcion. The ranlaional equaion of moion of he ball in he y direcion gie: F = m ( 1) ρ where i he radiu of curaure of he urn. The urn rae i relaed o he radiu of curaure wih: Ω = ρ ( ) Soling for in Equaion and ubiuing ino Equaion 1 gie he following expreion relaing he urning force o he urn rae: F = m Ω ( 3) The momen acing on he ball due o he reacion force (from he original diagram, on page 1) and he pin-down fricion orque (from he diagram aboe, on page 3) i: M T in F co iˆ mg mgdˆj T co F in r kˆ ( 4) The angular momenum of he ball i: H I m ˆ j k ˆ ( ) The rae of change of he angular momenum, relaie o he frame i: H rel a m ˆ j kˆ ( 6) where i he Englih pin-down rae of he ball. Noe ha here i no x componen in hi equaion becaue I auming for now ha no mae pin deelop.

6 The rae of change of angular momenum due o roaion of he frame i: ( 7) H mˆ i The equaion of moion for he ball' roaion, relaie o he roaing frame i: M d d H H rel H where he ecor are gien by Equaion 4, 6, and 7. ( 8) The y componen of Equaion 8 i he ame a Equaion 6, which implie he roll deceleraion rae i no affeced by he idepin. The z componen of Equaion 8 i imilar o Equaion 18, and i can be ued o find he Englih pin-down rae for he moing ball. The x componen of Equaion 8 can be ued o calculae he prediced ball "urn" rae: T in F co m Uing Equaion 3, we can now ole for he ball urn rae: T in m co If mae pin were included in he analyi, Equaion 30 would become: T in m m m co ( 9) ( 30) ( 31) where α m i he rae of change of mae pin. From hi equaion, if mae pin deelop and laer degrade (i.e. α m would change ign and magniude), he ball could urn in eiher direcion or go raigh (i.e., Ω could be poiie, negaie, and een zero during he moion), depending upon he relaie magniude of he mae and pin-down-orque erm in he numeraor. A decribed aboe (below Equaion 19), we will aume no mae pin deelop. eurning o Equaion 30, he ball urn rae i a funcion of ball peed: Ω ( ) 1 co( θ) T in( θ) m ( 3)

7 From Equaion, he ball' deceleraion rae i: a μ r g ( 33) For an iniial elociy o, he elociy of he ball a a funcion of ime i: o a o ( 34) and he ime o reach a cerain diance i: o x o o a x ( 3) a Therefore, he amoun of ball urn for a ho wih iniial peed o and rael diance L i: θ o L o L Ω o d ( 36) 0 and he ho error (laeral diance from he arge) would be: E o L L θ o x 0 dx ( 37) Here are ome example number for a long, low ho (wor cae) and a faer, horer ho: o mph L 8f 0.30deg θ o L E o L o L in o mph L 3f o L deg θ o L E o L in Obiouly, he "ball urn" effec i ery mall, a i expeced from oberaion (e.g., NV B.7), bu he phyic doe eem o ugge ha a low rolling ball wih idepin migh end o urn a ligh amoun in he direcion of he pin (e.g., righ pin caue urn o he righ).

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