TP A.31 The physics of squirt

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Cory Willis
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1 thnial proof TP A. Th physis of squirt supporting: Th Illustratd Prinipls of Pool and Billiards y David G. Aliator, PhD, PE ("Dr. Dav") thnial proof originally postd: 8//7 last rvision: /9/8 For a or dtaild, and slightly diffrnt, ovrag of squirt physis, s on Shpard's papr: "Evrything you always wantd to know aout u all squirt, ut wr afraid to ask." For or illustrations and asi xplanations, s y August '7 instrutional artil. F S C y C v x F ~ : forward ipuls ating on th all S ~ : sid (squirt) ipuls ating on th all : squirt angl C: point of ontat v: all spd aftr ipat : all ass : all angular spd aftr ipat : shaft nd fftiv ass ("ndass") : all radius : tip offst (Not - this is diffrnt fro Shpard's "")

2 Applying linar ipuls and ontu prinipls to th all in th x and y dirtions gi: whr t ~ ( ) F F( t) dt v t ~ ( ) S S( t) dt os s sin Not that fors F and S ar oponnts rsulting fro oth noral and frition fors ating during th rif ontat ti t. Baus th tip grips th all during ontat, th frition for is diffiult to odl dirtly. Applying linar ipuls and ontu prinipls to th shaft in th y dirtion gi: S ~ v C y ( 4) whr is th fftiv ass of th nd of th shaft (i.., th "ndass"), and v Cy is th spd of th point of ontat in th y dirtion. Equations and 4 suggst onsrvation of ontu in th vrtial dirtion. Th squirt ontu gaind y th all is aland y u stik dfltion in th opposit dirtion. Th fftiv ass is a funtion of gotry and atrial proprtis of th nd of th shaft. It rlats to how far th transvrs lasti wav travls down th shaft (fro th tip) during th rif ontat ti twn th tip and all. Exprints ha v suggstd that only th last 6 inhs or so of th shaft (losst to th tip) ontriut to nd ass. This is why u anufaturs hav n sussful with rduing squirt y using a sallr tip and shaft diatr, using a sallr and lightr frrul, and drilling out th nd of th shaft, all to rdu th fftiv ndass. It is iportant to not that Equation 4 assus th tip grips th all. Th tip is assud to rain in ontat with th all as th all rotats (s HSV A.76a for visual vidn of how wll a halkd lathr tip "gras" th all). Whil th tip and all ar in ontat, th vloity of th tip and all ar qual at th point of ontat. Th vloity of ontat point C, at th nd of th ipat priod, an writtn as: v C v v iˆ s ˆj kˆ iˆ s ˆj ( ) whr, fro th triangl in th figur aov, os( ) s sin( ) 6 so, fro Equations and 6, th vrtial (y or j) oponnt of th point-of-ontat vloity is: vc y 7

3 Applying angular ipuls and ontu prinipls to th all gi: I S F ~ ~ 8 whr I is th ass ont of inrtia of th all (/ ). Sustituting Equation 7 into Equation 4 and quating this to Equation gi: 9 Solving this quation for th u all angular spd gi: Sustituting Equations,, and into Equation 8 gi: v This quation an rwrittn as: s tan Thrfor, using Equation 6, th squirt angl an rlatd to th tip offst and shaft ndass as: tan

4 Blow is a plot of squirt angl (in rs). offst fator ( r = /) for th full rang of possil tip offsts ( = to.) and for a typial rang of all-ass-to-ndass ratios ( r = / ), ovring vrything fro a rak u (larg ndass and sall ass ratio) to a playing u with a low-squirt shaft (sall ndass and larg ass ratio). atan α r r r r r r r.. 4 α r α r α r r Th squirt angl is vry los to a linar funtion of tip offst (i.., th largr th offst, th largr th squirt, y a proportional aount).

5 Blow is a plot of squirt angl (in rs). all-ass-to-ndass ratios ( r = / ) for various aount of English (%, %, %). r. α %. r α %. r α %. r 4 4 r Hr is a plot of squirt (rs). ndass (gras) for diffrnt aounts of English: 6oz _low _high _low. _low _high α %. α %. α %. 4 g

6 So it appars that squirt varis fairly linarly with fftiv ndass, so a rtain prntag hang in ndass will rat los to th sa prntag hang in squirt. Hr is so xapl data for a % dras in ndass: high-squirt u: r α. α. r low-squirt u: r 4 r α. r.76 α. r r α r α. α..7 r α. r r α %. α. α. r r.7.4 r α. α..7 r α. r.74%

TP B.1 Squirt angle, pivot length, and tip shape

technical proof TP B.1 Squirt angle, pivot length, and tip shape supporting: The Illustrated Principles of Pool and Billiards http://illiards.colostate.edu y David G. Alciatore, PhD, PE ("Dr. Dave") technical