RES COVER APR QQ April 4th Final - ias

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378 RESONANCE April 2008

Classroom

In this section of Resonance, we invite readers to pose questions likely to be raised in a

classroom situation. We may suggest strategies for dealing with them, or invite responses,

or both. “Classroom” is equally a forum for raising broader issues and sharing personal

experiences and viewpoints on matters related to teaching and learning science.

Spinning Ball Flight Under Moderate WindK R Y Simha

Mechanical Engineering

Department

Indian Institute of Science

Bangalore 560 012, India.

Email:

[email protected]

Dhruv C Hoysall

Indian Institute of Technology

Madras

Chennai 600 013, India.

Email:

[email protected]

Keywords

Magnus effect; cycloid, spiroid,

terminal velocity, sine/cosine

integral, leg spin, off spin, s-

spin, top spin, side spin, back-

ward spin.

P r e d ic t in g t h e ° ig h t o f s p in n in g b a lls is a n e x -

c it in g a s p e c t w h ile p la y in g c r ic k e t , t e n n is , t a b le

t e n n is o r s o c c e r . T h e s e g a m e s d e m a n d a w id e

r a n g e o f s k ills t o e x p lo it t h e a e r o d y n a m ic e ® e c t s

in d u c e d b y s p in , w in d a n d g r a v it y . T h e t h e -

o r y u n d e r ly in g t h e s e a e r o d y n a m ic e ® e c t s u n v e ils

b iz a r r e o p p o r t u n it ie s fo r b u d d in g c r ic k e t , t e n n is

a n d s o c c e r s t a r s !

1 . I n t r o d u c t io n

A p re v io u s c la ssro o m a rticle [1 ] d iscu sse d th e e ® e c t o fstro n g w in d o n ten n is b a ll ° ig h t. H o w ev e r, m o re c o m -m o n ly, m o d e ra te w in d s p re v a il d u rin g p la y tim e . A lsob a d sh o ts h it b y b a tsm e n o ® th e e d g e o f th e b a t in d u cesp in to th e b a ll. T h e c o m b in e d e ® e c t o f sp in , d ra g a n d

g ra v ity p ro d u c e sp e cta cu la r e ® e c ts in te n n is b a ll c rick e ta n d so c ce r. A s d isc u sse d in [1 ], th e g o v e rn in g d i® e re n tia le q u a tio n s o f m o tio n u n d e r m o d e ra te w in d c o n d itio n a rea fo rm id a b le set o f co u p le d n o n -lin e a r d i® e re n tia l e q u a -tio n s! In g e n e ra l, th e n o n lin ea r e q u a tio n s d e fy a n a ly tic a l

stra te g ie s a n d d e m a n d n u m erica l m e th o d s fo r th eir so -lu tio n s. In th is a rtic le, w e re v isit th ese e q u a tio n s, a n d

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379RESONANCE April 2008

re -ex a m in e th e sp e cia l c a se o f th e b a ll re tu rn in g to th ep o in t o f p ro je ctio n . F o r th e sa k e o f c o n tin u ity, w e re -c a p tu la te th e m a in id ea s fro m th e p rev io u s c la ssro o m

a rticle o n ten n is b a ll ° ig h t w ith o u t sp in u n d er stro n gw in d .

2 . F lig h t W it h o u t S p in : S t r o n g W in d s

S u p p o se a b a tsm a n h its a ten n is b a ll. L et th e b a ll v e -lo c ity b e v b g , w h e re su p e rsc rip ts d en o te th e v e lo c ity o fth e b a ll w ith re sp e c t to th e g ro u n d . A ssu m in g a ste a d yw in d v e lo c ity v w g , th e rela tiv e v e lo c ity o f th e b a ll w ith

re sp e c t to th e w in d is v b w = v b g ¡ v w g . In g e n era l, v b g

a n d v b w h a v e a ll th re e c o m p o n e n ts w h ile v w g h a s o n lytw o co m p o n e n ts p a ra lle l to th e c rick e t g ro u n d . T h e d ra gfo rc e v a rie s a s th e sq u a re o f th e re la tiv e v e lo c ity m a g -n itu d e (v b w )2 a n d a c ts in th e d ire c tio n o p p o site to th e

v e c to r v b w . In v ec to r n o ta tio n , th e d ra g ex p re sse d p e ru n it m a ss is

D

m= ¡ k (v b w )2 v b w

v b w= ¡ k v b w v b w : (1 )

In e q u a tio n (1 ) k is th e d ra g c o e ± c ie n t, w h ich d ep e n d so n th e sh a p e , siz e , o rie n ta tio n , sp e ed a n d te x tu re (fu r,m o istu re, e tc.,) o f th e ° y in g o b je ct. It is c o n v e n ien t toin tro d u c e a sta n d a rd w in d sp ee d c o , w h ich p ro d u ce s ad ra g eq u a l to th e b a ll w e ig h t. F o r a sta n d a rd 6 0 g te n n isb a ll, c o is a b o u t 2 5 m / s. T h u s,

D = ¡ m gv b w

c 2o

v b w :

W e a re n o w w e ll a rm e d to a tta ck th e v e cto r eq u a tio n o fm o tio n :

¡ m a b g = (D + m g ); (2 )

w h e re a b g is th e a c c ele ra tio n v e c to r. In o rd e r to ¯ x th ed irec tio n s o f th e u n it v e cto rs, w e ta k e th e b a tsm a n a sth e o rig in o f c o -o rd in a tes (X ,Y ,Z ). T h e X -a x is is a lo n g

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th e p itch , Y -a x is is p o in tin g in th e d ire c tio n o f th e le gu m p ire , a n d Z -a x is p o in tin g to th e sk y a b o v e th e b a ts-m a n . T h e w in d v e cto r is v w g = (iv w g

x + jv w gy ) a n d th e

a c c ele ra tio n d u e to g ra v ity is { k g . T h e v ec to r e q u a tio no f m o tio n lo o k s d ec e p tiv ely sh o rt a n d sim p le , b u t w a itu n til y o u rea d a b o u t a ll th e c o m p lic a tio n s ca u se d b y th eD te rm ! E x p a n d in g th is eq u a tio n in to its sc a la r c o m p o -n e n ts.

a b gx = ¡ g

v b wx v b w

c 20

;

a b gy = ¡ g

v b wy v b w

c 20

;

a b gz = ¡ g

"

1 +v b w

z v b w

c 20

#

: (3 )

C lea rly , th e re is n o h o p e fo r a n e a sy so lu tio n c o n sid e rin gth a t h a rd ly a n y th in g is k n o w n a b o u t th e v a ria tio n o f a b g

a n d D w ith tim e in th e th re e-d im e n sio n a l, w in d -b lo w nsp a ce . It se e m s lik e it is n o t g o o d c rick e t! B u t, w a it!

F o r stro n g w in d s v b w ¼ ¡ v w g ! T en n is b a ll c rick e t d o e sn o t sto p b ec a u se o f a g a le o r tw o ! N o w , u n d e r th is g a lefo rc e , th e fo rm id a b le n o n lin ea r c o u p led e q u a tio n s b o wd o w n to sim p le d e c o u p led lin e a r e q u a tio n s:

a b gx = ¡ g

v b wx v b w

c 20

;

a b gy = ¡ g

v b wy v b w

c 20

;

a b gz = ¡ g : (4 )

It is in d e e d rem a rk a b le th a t stro n g w in d s m a k e th e g o -in g sm o o th b y w a y o f d e co u p lin g th e m a ze o f c o u p le dn o n -lin e a r d i® e re n tia l e q u a tio n s in to a d o cile se t o f th reelin e a r eq u a tio n s! T o k e ep th in g s e v e n m o re sim p le , w e

a ssu m e th a t th e w in d is b lo w in g a lo n g th e d ire ctio n o fth e p itch in to th e b a tsm a n a lo n g th e n eg a tiv e X -a x is.F u rth er, w e a ssu m e th e b a ll is h it e ith e r h ig h o v er th e

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Fine leg

Longstop

Third man

Slipe

WK

Gully

Point

Cover

Squareleg

SqLU

Mid-wicketU

BowlerMid-off Mid-on

Long off Long on

On (Leg) sideOff side

S

NS

Y

X

2e

Box 1. Cricket: Fielding Positions for a Right-handed Batsman.

b o w le r o r p u lle d lo w o v er th e sq u a re (see B o x 1 ) le g u m -p ire S q L U . In th e ¯ rst c a se th e b a ll b a llo o n s u p o v e rth e b o w le r a n d th e w in d d ra g s it b a ck to w a rd s th e b a ts-

m a n . C u rio u s? L e t u s d e riv e a fo rm u la to d riv e h o m eth is id e a . T h e d o c ile se t o f th re e eq u a tio n s b e c o m es asw e et se t o f tw o :

a b gx = ¡ g

µv w g

x

c 0

¶2

;

a b gz = ¡ g :

This map shows the

batsman (S), straight

umpire (U), square leg

umpire (Sq L U), and

fielder positions. The tussle

between the bowler and the

batter is orchestrated by the

captain and his team of

players positioned stra-

tegically in the oval field.

The ellipse with major axis

2a has the batter and the

bowler at the foci separated

by 2e. The polar equation

of the ellipse using the

batter as the origin is r =

p/(1– cos), where p is

the boundary distance

along Y-axis and is the

ratio e/a. Tennis ball

cricket, however, bends

geometric rules to blend in

with the available space.

The sprawling city of Bangalore and suburbs can perhaps boast of the largest tennis ball cricket following in

the world with well over a million players. This revolution was fuelled in the 1960s with Test cricket legends

like B S Chandrashekar and G R Vishwanath participating actively in suburban tournaments. There is no doubt

that tennis ball cricket represents the most imaginative Indian innovation in sports and pastime in recent times.

(Adapted from http://en.wikipedia.org/wiki/fielder)

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Figure 1. Flight paths in

vertical plane under strong

wind blowing left.

X

Z

O m ittin g su p e rsc rip ts a n d in teg ra tin g th e e q u a tio n s

v x = v x o ¡ ® g t;

v z = v z o ¡ g t;

w h e re ® = [v b w

x

c 0]2 ; a n d , v z o , v x o a re th e in itia l v e lo c ity

c o m p o n en ts. In teg ra tin g a g a in a n d elim in a tin g th e tim ev a ria b le t g iv es th e ° ig h t p a th . T h is w a s p ro v e d to b eg iv e n b y th e e q u a tio n

(x ¡ ® z )2

z v x 0 ¡ x v z 0

=2 (® v z 0 ¡ v x 0 )

g: (5 )

T h is e q u a tio n rep re se n ts a tilte d p a ra b o la . T h e a n g le o f

tilt is g iv e n b y a rc ta n (1 / ® ) w ith re sp ec t to th e g ro u n d .T h e w in d e ® e c t is lik e p la y in g o n a m o u n ta in slo p e.T h u s, stro n g w in d s a n d le v el p la y in g g ro u n d s d o n o tg o to g eth e r! S o m e ty p ica l tra jec to rie s a re sh o w n in F ig-u re 1 . O b serv e th e stra n g e b u t sp ec ia l c a se o f th e b a llm o v in g u p a n d d o w n a stra ig h t lin e . In th is ex tra o r-

d in a ry situ a tio n th e w in d re tu rn s th e b a ll b a ck to th eb a t. T h is is n o t a s o d d a s it se e m s w h e n th ere is n ow in d . A b a ll th ro w n v ertica lly u p co m e s d o w n to th esa m e p o in t e v e n tu a lly . U n d er lig h t b re ez e c o n d itio n s,th is sp e c ia l a n g le is a b o u t 8 0 d e g re es, a n d u n d er stro n g

w in d s a n g le s a s lo w a s 4 5 to 6 0 d eg ree s a re p o ssib le . U n -d e r m o d e ra te b ree z e, h o w e v e r th e b a ll e x ec u te s a b re e zylo o p re sem b lin g a n a irfo il sh a p e .

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Figure 2. Aerial view of

swinging ball under strong

wind blowing left.

X

Y

W e co n c lu d e d th e p re v io u s c la ssro o m a rtic le [1 ] w ith ase co n d e x a m p le o f a te n n is b a ll h it lo w o v e r th e sq u a rele g u m p ire . W e a re n o w in te re ste d in th e w a y th e b a ll

d rifts in th e d ire c tio n o f th e w in d w h e n v ie w e d fro mth e to p in th e X Y p la n e. T h ere fo re , w e n ee d o n ly tw oe q u a tio n s.

a b gx = ¡ g

µv w g

x

c 0

¶2

;

a b gy = 0 :

In te g ra tin g a fte r d ro p p in g su p erscrip ts, w e g e t

v x = ® g t;

v y = v y 0 = c o n st:

In th is ca se , th e w in d c a rrie s th e b a ll b eh in d th e p o si-tio n o f th e le g u m p ire b e fo re th e b a ll h its th e g ro u n d .L o o k in g fro m a b o v e , th e b a ll sw e ep s a p a ra b o lic a rc in

th e X Y -p la n e (F igu re 2 ), g iv e n a s fo llo w s

y 2 +

Ã2 v 2

y o

® g

!

x = 0 ; (6 )

3 . S p in n in g B a ll F lig h t : M o d e r a t e W in d s

In sp ire d b y th e le g e n d a ry fre e k ick ex e c u ted b y B e ck h a mo n 2 6 th J u n e 1 9 9 8 , C o o k a n d G o ® [2 ] e x p lo red th e c o m -b in a tio n o f p a ra m e ters fo r su c c essfu l so c ce r k ick s. T h eir

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The magnus force is

directly proportional to

the cross product of

spin(s) and relative

velocity of the ball

with respect to air (R).

c a lcu la tio n s su g g est th a t B eck h a m k ick ed th e so c c er b a lla t a b o u t 1 0 0 k m / h w ith a sp in o f 6 0 0 rp m ! T h is k ickro se w ell o v e r a b e w ild e re d w a ll o f d e fen d ers e n ro u te

th e to p ed g e o f th e g o a l ( 6 0 ft y o n d e r, 8 ft u p a n d1 2 ft to th e le ft). T h e re is a tra d e o ® b etw e en sp ina n d tra n sla tio n w h ile b en d in g th e so c c er b a ll in ° ig h t.In ten n is b a ll c rick e t, h o w e v e r, sp in n in g sh o ts o c cu r b ych a n ce ra th e r th a n b y d esig n . T h e a d d itio n a l sp in in -

d u c ed a ero d y n a m ic fo rc e is th e fa m o u s m a g n u s e ® e ct.T h e m a g n u s fo rc e is d ire ctly p ro p o rtio n a l to th e c ro ssp ro d u c t o f sp in (s ) a n d re la tiv e v e lo c ity o f th e b a ll w ithre sp e c t to a ir (R ). T h e co n sta n t o f p ro p o rtio n a lity k s ,lik e th e d ra g c o e± c ie n t, d ep e n d s o n th e sh a p e , size a n dte x tu re o f th e b a ll. T h is a d d itio n a l a e ro d y n a m ic fo rce

g iv e s th e n e w eq u a tio n o f m o tio n ,

m a b g = (D + m g ) + k s s £ R : (7 )

D ra g fo rc e is p ro p o rtio n a l to th e sq u a re o f re la tiv e v e lo c -ity a n d a c ts o p p o site in d ire c tio n to th e re la tiv e v e lo c ityv e c to r. W e ta k e c w a s th e w in d sp e ed a n d w h e n th e w in dsp ee d a tta in s c o , th e d ra g b ec o m e s e q u a l to th e w eig h to f th e b a ll. T h u s, c o is a lso th e te rm in a l v e lo c ity o f th e

b a ll fa llin g fre e ly in th e v e rtica l d ire ctio n . W ith re sp ec tto th e sp in n in g b a ll, a sp in o f m a g n itu d e s o a n d w in dsp ee d c o p ro d u c e a m a g n u s fo rc e o f m g . B a sed o n C o o ka n d G o ® d a ta fo r so c c er b a ll[2 ], c o is a b o u t 2 5 m / s a n ds o a b o u t 6 0 0 rp m . It is ra th er a re m a rk a b le c o in cid e n ce

th a t c o fo r a so cc e r b a ll is a b o u t th e sa m e a s fo r a te n n isb a ll!

T h e b a tsm a n is a t th e o rig in o f c o o rd in a te s (X ,Y ,Z );X -a x is is a lo n g th e p itch a n d w in d is b lo w in g in th e

n e g a tiv e X d ire ctio n (i.e ., to w a rd s th e b a tsm a n ). W eta k e Y -a x is in th e v ertic a l d ire ctio n in th e se q u el; a n d ,th e refo re, th e Z -a x is is n o w p o in tin g a w a y fro m th e le gu m p ire . H o w e v e r, w e d o n o t re a lly n ee d a n y sp e c i cv a lu e fo r c 0 if w e u se n o n -d im e n sio n a l p a ra m e ters. T h e

re la tiv e v elo city o f th e b a ll w ith resp e ct to th e w in d is

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d e n o te d a s R = [(u + c w )2

+ v 2 + w 2 ]1 = 2 , w h e re u ; v ; w

re p resen t th e b a ll v elo city c o m p o n en ts. T h e g o v e rn in ge q u a tio n s o f m o tio n o f a sp in n in g b a ll o f m a ss m a re :

md u

d t= ¡ k d (u + c w )R + k s (s 2 w ¡ s 3 v );

md v

d t= ¡ k d v R ¡ m g + k s (s 3 (u + c w ) ¡ s 1 w );

md w

d t= ¡ k d w R + k s (s 1 v ¡ s 2 (u + c w )): (8 )

In th e a b o v e e q u a tio n s s 1 , s 2 , s 3 a re th e sp in co m p o n e n tsa b o u t X , Y a n d Z a x e s, re sp e c tiv e ly. It is c o n v e n ien tto n o rm a lize th e d ra g a n d m a g n u s fo rce w ith re sp ec tto th e w e ig h t o f th e b a ll b y n o rm a liz in g th e v e lo c ityc o m p o n en ts w ith resp e ct to c o a n d sp in c o m p o n en ts w ith

re sp e c t to s o . T h e n o n -d im e n sio n a l fo rm o f e q u a tio n s o fa sp in n in g b a ll b e c o m es

1

g

d u

d t= ¡ (u + c w )R = c o

2 + (s 2 w ¡ s 3 v )= c o s o ;

1

g

d v

d t= ¡ (v R + c o

2 )= c o2 + (s 3 (u + c w ) ¡ s 1 w )= c o s o ;

1

g

d w

d t= ¡ (w R )= c o

2 + (s 1 v ¡ s 2 (u + c w ))= c o s o : (9 )

T h e a b o v e se t re se m b le s V o lte rra { L o tk a eq u a tio n s u se dfo r a tm o sp h eric a n d e c o lo g ic a l m o d e llin g . T h ese e q u a -

tio n s a re o fte n e x p re ssed in th e K o lm o g o ro v fo rm :

d u

d t= u f (u ; v ; w );

d v

d t= v g (u ; v ; w );

d w

d t= w h (u ; v ; w ):

L o ren z d e v elo p ed sim ila r eq u a tio n s to o p e n th e d o o rs o fn o n lin ea r d y n a m ic s a n d ch a o s [4 ]. T h e g e n e ra l ca se o fth e sp in n in g b a ll ° ig h t re q u ire s 4 in itia l v e lo c ities (u o ; v o ;

w o ; c w ) a n d th re e sp in c o m p o n en ts (s 1 ; s 2 ; s 3 ). W e a s-

su m e in th is fo rm u la tio n th a t th e re is n o d ec a y in sp inw ith tim e (js j = c o n sta n t).

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Figure 3. Flight paths for

increasingprojectionangle

(gentle wind blowing left).

0

20

40

60

80

100

120

80

-600

20

40

60

200 40 60 80 200 40 60 80 -40 -20 60400 20

40

0

20

60

80

120

100

140

100

120

Y

X

Y Y

X X

4 . S p in le s s F lig h t : M o d e r a t e W in d s

In th is sec tio n , w e re co n sid er th e ° ig h t w ith o u t sp in inth e v e rtic a l (X Y ) p la n e. O n c e a g a in , th e id e a is to e x -p lo re th e p o ssib ility o f th e b a ll retu rn in g to th e o rig info r a g iv en set o f v a lu es (u o ; v o ; c w ). U n lik e th e te n -n is b a ll ° ig h t u n d e r stro n g w in d , ° ig h t u n d e r m o d e ra tew in d o ® ers lim ited th e o re tica l sc o p e fo r d eriv in g c lo se d

fo rm so lu tio n s. It b e c o m e s n e cc e ssa ry to ex p lo re n u m e r-ic a l stra te g ie s to g a in in sig h t in to th e ° ig h t o f th e te n n isb a ll. F o r th e sa k e o f p ro v id in g a p h y sic a l fee l fo r th ep ro b le m , w e illu stra te th e rich v a rie ty o f ° ig h t p a th s b ya ssu m in g c w = 3 0 m / s , c o = 4 4 :3 m / s, R o = 1 0 0 m / s

in F igu re 3 .

T h e re is a sp ec ta c u la r d iv e rsity o f ° ig h t p a th s p o ssib led e p e n d in g o n th e in itia l a n g le o f p ro jec tio n (F igu re 3 ).

O b se rv e th e c a se o f a lo o p in g ° ig h t p a th . T h is lo o pg ra d u a lly e v o lv es in to a c u sp . A t a sp e c ia l a n g le o f p ro -jec tio n , th e te n n is b a ll re tu rn s to th e o rig in . A b o v e th issp ec ia l a n g le th e te n n is b a ll fa lls b eh in d th e b a tsm a n(a n d h o p efu lly n o t in to th e w ick etk ee p e rs h a n d s!).

T o ex p re ss re su lts u sin g n o n -d im en sio n a l q u a n titie s, w ec a n ta k e c o a s th e u n it o f sp e ed . W e d e ¯ n e n o n -d im e n -

sio n a l v e lo c ity r = (u 2 + v 2 )1 = 2

. T h e u n it o f le n g th is

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Figure 4. Return Paths [ ro=

0.226, o

= 64.2o, 43.5o,

10.75o ], [ ro= 2.26,

o= 56.3o,

38.7o, 9.69o ], [ ro

= 22.6, o=

52.03o, 37.2o, 9.64o].

w

0.30.1 0.2010.90.80.70.60.50.40.30.20 0.1

C =2.26

C =.68

C =1

w

w

w

0.3

C =2.26w

0.1

0

0.2

0.5

0.4

0.6

10.90.80.70.60.50.4

C =2.26w

C =1w

C =1

C =.68w

0.8

0.7

1

0.9

C =.68w

0

0.1

0.4

0.2

0.3

0.6

0.5

0.7

0.9

0.8

1

0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1X X

X

Y

Y Y

ta k en a s h o , th e h e ig h t a tta in e d b y th e b a ll w h e n th ro w nv e rtic a lly w ith a v elo city R o a ssu m in g a n e® ec tiv e v a lu eo f g ra v ity g ¤ = g (sin µ o + ® co s µ o ), w h e re µ o is th e a n g le

o f p ro jec tio n a n d ® = c w2 = c o

2 . W h en th ere is n o w in d

(c w = 0 ), it c a n b e sh o w n th a t h o = c o2

2 g ¤ ln (1 + R o2

c o2 ), fo r

v e rtic a l ° ig h t.

F igu re 4 p resen ts n o n -d im e n sio n a l tra jec to rie s fo r th reed i® e ren t n o n -d im e n sio n a l w in d sp e e d s c w = 0 :6 8 ; 1 :0 ; 2 :2 6 .It is in te re stin g to o b se rv e th e lo o p s g ettin g n a rro w e r

fo r d e c rea sin g v a lu e s o f r o . F o r v a n ish in g ly sm a ll r o ,th e sp e c ia l a n g le µ s = ¼ = 2 ¡ ® a s d isc u sse d in [1 ] fo rstro n g w in d s. T h e re tu rn p a th a lig n s w ith th e te rm in a lv e lo c ity v e c to r. T h e term in a l v e lo c ity v e c to r is v e rtic a lw h e n th e re is n o w in d . T h e c o n ce p t o f term in a l v e lo c -

ity in ste a d y h e a d w in d im p lies th a t th e b a ll is ca rrie d

CLASSROOM


Figure 5. Spinning ball re-

turning to origin; uo= v

o= –

wo= .0226c

o; c

w= c

o= s

2.

b y th e w in d (u = ¡ c w ). T h e v e rtic a l c o m p o n en t o f th ete rm in a l v elo city is c o . T h ere fo re , th e te rm in a l a n g le µ t

sa tis¯ e s ta n µ t = (c o = c w ). T h e a n g les µ s , µ t a n d th e in itia l

a n g le o f p ro je c tio n µ o co m p le te th e k in em a tic p ic tu re o fth e ° ig h t.

5 . S p in n in g B a ll: S t r o n g W in d s

It is in tere stin g to stu d y th e ° ig h t o f sp in n in g b a ll u n d e rstro n g w in d b ec a u se th e e q u a tio n s sim p lify en o rm o u sly.U n d er stro n g w in d u ; v ; w < < c w ; a n d s 1 = s 3 = 0 ,

d u

d t= ¡ g

c w2

c o2

;

d v

d t= ¡ g ;

d w

d t= ¡ g

s 2 c w

c o s o: (1 0 )

T h is c a se is sim ila r to th e te n n is b a ll ° ig h t u n d e r stro n gw in d ex c ep t th a t th e p a th o f th e b a ll is n o t in th e X Y -p la n e (se e F igu re 5 fo r th e sp ec ia l c a se o f b a ll re tu rn in gto th e o rig in ). T h e m o tio n o f th e b a ll w o u ld b e e x ec u te d

o n a stra ig h t lin e in c lin ed eq u a lly to th e co o rd in a te a x es,if u o = v o = ¡ w o = :0 2 2 6 c o ; c w = c o = s 2 . T h e re su lt issh o w n in th e to p a n d sid e v ie w s in F igu re 5 .

X

0.02-0.02 00

0.02

0.060.04 -0.02

0.06

0.04

0.040

0 0.02

0.02

0.06

0.06

0.04

X

Z Y

CLASSROOM


Figure 6. Spinning ball re-

turning to the origin: Mod-

erate winds.

Y

10

0

0

5 10

5

15

25

20

2015 25 3530 40

35

30

30

10

0

0 10 20

20

30 40

40

X X

Z

6 . S p in n in g B a ll: M o d e r a t e W in d s

W h e n th e b a ll a n d w in d sp e ed s a re o f th e sa m e o rd er,th e tra je cto ry o f th e b a ll is sh o w n in F igu re 6 . T h e b a lle x e c u tes a 3 -D lo o p b e fo re re tu rn in g to th e o rig in fo rth e c o n d itio n s u o = v o = ¡ w o = c o , c w = c o . T h is° ig h t tra je c to ry h a s c o n sta n tly ch a n g in g cu rv a tu re a n dto rsio n .

7 . S p in n in g B a ll: A e r ia l V ie w

W in d a n d g ra v ity c o m p lic a te m a tte rs. T h e re a re situ -a tio n s w h ere g ra v ity a n d w in d b e co m e re la tiv e ly u n im -p o rta n t. T h e a b se n ce o f g ra v ity a n d w in d , a s w e w ill se e,b rin g s o u t a rich d iv e rsity o f ° ig h t p a th s a s w ith frisb e e sa n d b o o m e ra n g s. L e t u s im a g in e th a t th e b a ll is in a

g ra v ity - a n d w in d -fre e en v iro n m en t. L e t u s th ro w th eb a ll sp in n in g a b o u t th e Y -a x is in th e X Z -p la n e a n d a lo n gth e X -a x is w ith v = w = 0 . L e t s 2 6= 0 ; s 3 = s 1 = 0 .T h e eq u a tio n s g o v e rn in g th e m o tio n o f th e b a ll red u ceto :

d u

d t= ¡

u R

c o2

g + gs 2 w

c o s o;

d w

d t= ¡

w R

c o2

g ¡ gs 2 u

c o s o

: (1 1 )

CLASSROOM


The absence of

gravity and wind as

we will see, brings

out a rich diversity

of flight paths as

with frisbees and

boomerangs.

P u ttin g u = R c o s µ a n d w = R sin µ , w e c a n sh o w th a t

d R

d t= ¡ g

R 2

c o2

:

U p o n in te g ra tio n w e g et

R (t) = R oc o

2

c o2 + R o g t

:

T h e co rresp o n d in g so lu tio n fo r µ (t) = ¡ g s 2

c o s ot is a s fo l-

lo w s

u = R o

c o2

(c o2 + R o g t)

c o s(¡g s 2

c o s o

t);

w = R oc o

2

(c o2 + R o g t)

sin (¡g s 2

c o s ot): (1 2 )

T a k in g a = (c o s 2 )= (R o s o ) a n d c = (g s 2 )= (c o s o ); th e ° ig h tp a th in th e X Z -p la n e is g iv e n in term s o f sin e a n d c o sin ein te g ra ls.

x (t) =c o

2

g((S i(a + c t) ¡ S i(a ))sin (a )

+ (C i(a + c t) ¡ C i(a ))co s(a ));

z (t) =c o

2

g((C i(a + c t) ¡ C i(a ))sin (a )

¡ (S i(a + c t) ¡ S i(a ))c o s(a )): (1 3 )

In th e a b o v e so lu tio n S i(x ) a n d C i(x ) a re th e d e ¯ n ite

in te g ra ls b e tw ee n th e lim its 0 to x o f th e fu n c tio n s sin (x )x

a n d c o s(x )x , resp e ctiv e ly. W e n o tic e th e g ra d u a l d e ca y

in sp e ed w ith tim e a s th e b a ll sp ira ls to w a rd s its ¯ n a lp o sitio n . T h is ¯ n a l p o sitio n d ep e n d s o n th e in itia l v e -

lo c ity a n d sp in . T h e sp ira l v a ria tio n o f th e v elo city isp o rtra y e d in F igu re 7 . N o te th a t th e v e lo c ity o f th esp in n in g b a ll v a ries in v ersely w ith tim e e la p se d v a n ish -in g ev e n tu a lly. T h e sp in n in g b a ll a p p ro a ch e s its ¯ n a ld e stin a tio n w ith c o o rd in a te s (6 8 .7 ,{ 1 2 4 .3 )m w h en th e

CLASSROOM


Figure 7. Spinning ball spi-

ral in XZ-plane and veloc-

ity plot for Ro= c

oand s

2=

so.

120

ZD

ista

nce

(m)

-80

-140

-180-20 0

-160

-120

-100

X Distance (m)

6020 40 80 100

-60

-40

-20

0SPIROD IN ABSENCE OF WIND

VV

eloc

ity

alon

gY

-0.1

U Velocity along X

-0.4

-0.5140 -0.2-0.4 0

-0.3

-0.2

0.60.2 0.4 0.8 1.0

PONCARE PLOT IN ABSENCE OF WIND

0

0.1

0.2

There is a dramatic

transformation in the

flight path in the

presence of a slight

tail wind behind the

spinning ball.

b a ll is th ro w n a t sp e e d u o = c o w ith sp in s 2 = s o . T h e se¯ g u re s a re sim ila r to sa te llite p ic tu re s o f cy c lo n es.

8 . T a il/ h e a d W in d E ® e c t : A e r ia l V ie w

T h e re is a d ra m a tic tra n sfo rm a tio n in th e ° ig h t p a th inth e p resen c e o f a slig h t ta il w in d b e h in d th e sp in n in g

b a ll. T h e v e lo c ity d ia g ra m is sh ifted b y c w , b u t o th e r-w ise a p p e a rs sim ila r to F igu re 7 . T h e re su ltin g ° ig h tp a th F igu re 8 , h o w ev er, is d ra stic a lly d i® ere n t. T h e ta ilw in d ta k es th e b a ll a lo n g a d isto rte d c y clo id . T h is p a th -c a lle d sp iro id , b len d in g sp ira l w ith th e c y clo id , d isp la y s

e x o tic fe a tu re s u n im a g in a b le w ith o u t sp in . D e p e n d in go n th e m a g n itu d e o f th e ta il w in d , th e ° ig h t p a th c a ne x h ib it b o th co n tra cte d a n d e x te n d e d ch a ra cte ristic s o fa c y clo id . T h is is c le a rly e v id e n t th ro u g h th e lo o p s a n dc u sp s in F igu re 8 . O b se rv e th e ¯ n a l o ® se t o f th e b a ll

fro m th e o rig in a l d ire c tio n o f ° ig h t (w h ich d e p e n d s o n lyw ea k ly o n w in d sp e ed ). T h ese re su lts a re a t ¯ rst c o u n te rin tu itiv e, b u t it sh o u ld b e n o te d th a t w e h a v e u se d ala rg e v a lu e o f s 2 = s o to d em o n stra te th e se e x o tic v a rie tyo f sp in n in g b a ll tra jec to rie s. T w o d i® e re n t m a g n itu d e s

fo r th e ta il w in d a n d o n e v a lu e fo r h e a d w in d illu stra te

CLASSROOM


-40

0

2500

-20

ZD

ista

nce

(m)

-120

-160

-180-1000

-140

-80

-100

-40

-60

X Distance (m)-800 -600 -200-400

0

-140

-120

0

ZD

ista

nce

(m)

-80

-100

-60

SPIROD UNDER HEAD WIND

X Distance (m)500 1000 20001500

200

40003000

-160

-180200100 300

ZD

ista

nce

(m)

-100

-120

-140

-60

-80

800

X Distance (m)

600500 700 1000900

-20

0SPIROD UNDER TAIL WIND SPIROD UNDER TAIL WIND

-20

-40

0

Figure 8. Aerial view of spinning ball

path: tail winds (co/10 and c

o/3) (top)

and head wind (co/10)(bottom). Note:

Ro= c

oand s

2= s

o.

h o w th e sp in n in g b a ll sn a k e s a lo n g u n til th e ta il/ h e a dw in d s d o m in a te th e d y n a m ic s.

In th e co n te x t o f te n n is b a ll c rick e t o r so c c er, a p la y e ra tte m p tin g to c a tch a sp in n in g a n d sn a k in g b a ll w illh a v e to b e a g ile a n d a le rt. T h is e ® e c t a lso m a k e s itp o ssib le fo r p itch e rs a n d b o w le rs to e x p lo it w in d a n dsp in to c o n fu se th e b a tter. S o , in ten n is b a ll c rick e t, in

a d d itio n to th e le g -sp in a n d o ® -sp in , w e h a v e th e sn a k in gs-sp in ! T h e se sp iro id a l p a th s a re a lso o b se rv e d d u rin gsw irlin g d u st sto rm s a n d c y c lo n e s.

9 . S u m m a r y

T h o u g h n o t ju st lim ited to so c c er a n d c rick e t, sp in is

CLASSROOM


Suggested Reading

[1] K R Y Simha, Tennis ball

flight under strong wind,

Resonance, Vol.7, No.11,

November 2002.

[2] B G Cook and J E Goff,

Parameter space for suc-

cessful soccer kicks, Eu-

ropean Journal of Phys-

ics, Vol. 27, pp.865–874,

2006.

[3] R D Lorenz, Spinning

Flight: Dynamics of

Frisbees, Boomerangs,

Samaras, and Skipping

Stones, Springer 2006.

[4] S H Strogatz, Nonlinear

Dynamics and Chaos,

Westview Press, p.301,

1995.

u se d b y p la y e rs to g a in a n a d v a n ta g e o v er th e ir o p p o -n e n ts. T h is m a k e s th e u n d ersta n d in g o f th e d y n a m ic so f th e sp in n in g b a ll ev e n m o re im p o rta n t a s it c o u ld

h e lp p la y ers im p ro v e th e ir g a m e . T h e e® ec ts o f sp in a rec le a rly se en in g a m e s lik e ten n is a n d ta b le te n n is (T T ).H e re th e p la y e r c a n im p a rt 3 k in d s o f sp in s to th e b a ll.

F o rw a rd sp in o r to p sp in : T T p la y e rs w ill k n o w h o w

d i± c u lt it is to k e ep th e b a ll o n th e sm a ll ta b le w h e nh ittin g a fa st sh o t. T h is p ro b le m is o v e rc o m e b y im -p a rtin g a to p sp in to b a ll w h ile h ittin g a fa st sh o t. T h eto p sp in c a u ses th e b a ll to d iv e , h e n ce su rp risin g th eo p p o n e n t a n d p itch in g b efo re h e e x p e c ts it.

B a ck w a rd sp in : T h e b a ck w a rd sp in im p a rte d b y a p la y e rc o u n ters g ra v ity (th e m a g n u s e ® e ct) a n d th is c a u ses th eb a ll to ° o a t in th e a ir lo n g er th a n ex p e c ted . In th is

w a y ev en a slo w sh o t c a n b e m a d e to cro ss th e n e t b yim p a rtin g a b a ck w a rd sp in to th e b a ll.

S id e sp in : S id e sp in is b est o b se rv e d d u rin g a se rv e inT T . W h en th e p la y er im p a rts a sid e sp in to th e b a ll th e

b a ll sw in g s sid e w a y s (m a g n u s e ® e ct) a n d m o v e s fu rth e ra w a y fro m th e o p p o n e n t, m a k in g it to u g h e r fo r h im tore ce iv e th e se rv e .

T h u s, a e ro d y n a m ic v iew o f sp o rts fu rn ish es a d ee p e ra p p re c ia tio n o f th e p la y e rs sk ills a n d p e rse v e ra n c e tom a ste r w in d , sp in a n d g ra v ity . S o fa r w e d iscu ssed th ed y n a m ic s o f sm o o th sp in n in g b a lls. B u t so m e b a lls (e .g .,g o lf b a lls) h a v e d im p le s o n th em , so m e a re n o t c irc u la r

(ru g b y b a lls); th e d y n a m ics o f th e se b a lls a re v e ry d i® e r-e n t a n d m u ch m o re ch a lle n g in g ! T h e in itia l in sp ira tio nfo r th is a rtic le ca m e fro m o b se rv in g th e sa m a ra se e d ssp in n in g w h ile fa llin g d o w n fro m th e tre e. P re se n tly,th e re is a te a m o f stu d e n ts d e sig n in g a pa ra co p ter b a se d

o n th e sp in n in g se e d p rin c ip le to a ch iev e a te rm in a l v e -lo c ity o f le ss th a n a b o u t 4 m / s fo r a 1 0 0 k g p a ck a g ed ro p p e d fro m a h e ig h t o f 1 0 0 m .

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