## Spinning Out

I don’t know why, but last week was my most popular week ever, at least in terms of blog hits! I was going to follow up with a foray into the role of spin in quantum mechanics, but decided instead to settle for a less ambitious project for this evening.

Yesterday I walked past the cricket ground at the SWALEC Stadium in Sophia Gardens, Cardiff, during the Twenty20 international between England and Pakistan. There is another match of this type tomorrow night which I’ll actually be going to, as long as it’s not rained off, but I have too many things to do to go to both games. Anyway, England’s excellent off-spinner Graham Swann was bowling when I watched through a gap in the stands at the river end of the stadium. He seemed to be getting an impressive amount of turn, and I got wondering about how fast a bowler like “Swannee” actual spins the ball.

For those of you not so familiar with cricket here’s a clip of another prodigious spinner of the ball, Australia’s legend of legspin Shane Warne:

For beginners, the game of cricket is a bit similar to baseball (insofar as it’s a game involving a bat and a ball), but the “strike zone” in cricket is a physical object ( a “wicket” made of wooden stumps with bails balanced on the top) unlike the baseball equivalent, which exists only in the mind of the umpire. The batsman must prevent the ball hitting the wicket and also try to score runs if he can. In contrast to baseball, however, he doesn’t have to score; he can elect to play a purely defensive shot or even not play any short at all if he judges the ball is going to miss, which is what happened to the hapless batsman in the clip.

You will see that Warne imparts considerable spin on the ball, which has the effect of making it change direction when it bounces.  The fact that the ball hits the playing surface before the batsman has a chance to play it introduces extra variables that you don’t see in baseball,  such as the state of the pitch (which generally deteriorates over the five days of a Test match, especially in the “rough” where bowlers have been running in). A spin bowler who causes the ball to deviate from right to left is called a legspin bowler, while one who makes it turn the other way is an offspin bowler. An orthodox legspinner generates most of the spin from a flick of the wrist while an offspinner mainly lets his fingers do the torquing.

Another difference that’s worth mentioning with respect to baseball is that the ball is bowled, i.e. the bowler’s arm is not supposed to bend during the delivery (although apparently that doesn’t apply if he’s from Sri Lanka). However, the bowler is allowed to take a run up, which will be quite short for a spin bowler, but long like a javelin thrower if it’s a fast bowler. Fast bowlers – who can bowl up to 95 mph (150 km/h) – don’t spin the ball to any degree but have other tricks up their sleeve I haven’t got time to go into here. A typical spin bowler delivers the ball at speeds ranging from 45 mph to 60 mph (70 km/hour to 100 km/hour).

The physical properties of a cricket ball are specified in the Laws of Cricket. It must be between 22.4 and 22.9 cm in circumference, i.e. 3.57 to 3.64 cm in radius and must weigh between 155.9g and 163g. It’s round, made of cork, and surrounded by a leather case with a stitched seam.

So now, after all that, I can give a back-of-the-envelope answer to the question I was wondering about on the way home. Looking at the video clip my initial impression was that the ball is deflected  by an angle as large as a radian, but in fact the foreshortening effect of the camera is quite deceptive. In fact the ball deviates by less than a metre between pitching and hitting the stumps. There is a gap of about 1 metre between the popping crease (where the batsman stands) and the stumps – it looks much less from the camera angle shown – and the ball probably pitches at least 2 metres in front of the crease. I would guess therefore that it actually deflects by an angle less than twenty degrees or so.

What happens physically is that some of the rotational kinetic energy of the ball is converted into translational kinetic energy associated with a component of the velocity  at right angles to the original direction of travel. In order for the deflection to be so large, the available rotational kinetic energy must be non-negligible compared to the original kinetic energy of the ball. Suppose the mass of the ball is $M$, the translational kinetic energy is $T=\frac{1}{2} Mv^2$ where $v$ is the speed of the ball. If the angular velocity of rotation is $\omega$ then the rotational kinetic energy $\Omega =\frac{1}{2} I \omega^2$, where $I$ is the moment of inertia of the ball.

Approximating the ball as a uniform sphere of mass $M$ and radius $a$, the moment of inertia is $I=\frac{2}{5}Ma^2$.  Putting $T=\Omega$, cancelling $M$ on both sides and ignoring the factor of $\frac{2}{5}$– because I’m lazy – we see that the rotational and translational kinetic energies are comparable if

$v^2 \simeq a^2\omega^2,$

or $\omega \simeq \frac{v}{a}$, which makes sense because $a\omega$ is just the speed of a point on the equator of the ball owing to the ball’s rotational motion. This equation therefore says that the speed of sideways motion of a point on the ball’s surface must be roughly comparable to speed of the ball’s forward motion. Taking $v=80$ km/h gives $v\simeq \frac{80 \times 10^3}{60 \times 60} \simeq 20$ m/s and $a\simeq 0.036$ m gives $\omega \simeq 600$ radians per second, which is about 100 revolutions per second. This would cause a huge deviation (about 45 degrees), but the real effect is rather smaller as I discussed above (see comments below). If the deflection is actually around 15 degrees then the rotation speed needed would be around 30 rev/s.

This estimate is obviously very rough because it ignores the direction of spin and the efficiency with the ball grips on the pitch – friction is obviously involved in the change of direction – but it gives a reasonable ballpark (or at least cricketground) estimate.

Of course if the bowler does the same thing every time it’s relatively easy for the batsman to allow for the spin. The best  bowlers therefore vary the amount and angle of spin they impart on each ball. Most, in fact,  have at least two qualitatively different types of ball but they disguise the differences in the act of delivery. Offspinners typically have an “arm ball” which doesn’t really spin but holds its line without appearing to be any different to their spinning delivery. Legspinners usually have a variety of alternative balls,  including a topspinner and/or a flipper and/or a googly. The latter is a ball that comes out of the back of the hand and actually spins the opposite way to a legspinner while being produced with apparently the same action. It’s very hard to bowl a googly accurately, but it’s a deadly thing when done right.

Another thing also worth mentioning is that the rotation of the cricket ball also causes a deviation of its flightpath through the air, by virtue of the Magnus effect. This causes the ball to curve in the air in the opposite direction to which it is going to deviate on bouncing, i.e. it would drift into a right-handed batsman before breaking away from him off the pitch. You can see a considerable amount of such movement in the video clip,  away from the left-hander in the air and then back into him off the pitch. Nature clearly likes to make things tough for batsmen!

With a number of secret weapons in his armoury the spin bowler can be a formidable opponent, a fact that has apparently been known to poets, philosophers and astronomers for the best part of a thousand years:

The Ball no Question makes of Ayes and Noes,
But Right or Left, as strikes the Player goes;
And he that toss’d Thee down into the Field,
He knows about it all — He knows — HE knows!

The Rubaiyat of Omar Khayyam [50]

### 16 Responses to “Spinning Out”

1. Anton Garrett Says:

The recently retired Cambirdge University, then Kent then Middlesex – and in 2003 England – batsman Ed Smith was astute enough to do some pre-season training with a US baseball team and then write a book, Playing Hard Ball, which compares the two games and would be expected to sell on both sides of the pond.

I’ve seen the spin rate quoted from experimental determinations and from (fallible) memory you are about one order of magnitude too great. I’ve not thought why. It also emerged that wrist spinners could give it twice the rate of rotation as finger spinners. I’ll try and hunt down the reference.

Anton

2. The estimate I’ve quoted could certainly be wrong by a significant factor, but I doubt if it’s as larger as a factor of 10. For one thing there is a dependence on the angle with which the ball hits the ground relative to the direction of spin. Also the ball will slow down owing to air resistance before it reaches the point of bounce…

I have seen papers talking about experiments that reproduce swing bowling that include spin rates of about 11-14 rev/s. I’m sure the initial spin rate generated by the likes of Warne must be much bigger than that. A quick google didn’t turn up any specific numbers, but apparently baseballs spin at about 30 rev/s and golfballs at about 100 rev/s, which is enough to make them spin backwards.

3. Anton Garrett Says:

If you estimate 100 revs/s and the experiments find 11-14 then that is about one order of magnitude?

I think Warne is exceptional in his combination of large tweak and accuracy. There are other people around who can tweak it like that but you try putting it on a sixpence as well.

I’m still googling for the rotation rates of wrist and finger spinnners…

• Yes, but that’s a swing bowler. They don’t tweak it at all. Incidentally, about 10 rev/s is also what footballers put on free kicks too.

4. Anton Garrett Says:

Oops, I missed the word ‘swing’ in yours of 10:59pm. But I recently tracked down the “greatest free kick goal of all time” on YouTube (by Roberto Carlos for Brazil vs France in 1997) as a result of this item on the BBC website:

http://www.bbc.co.uk/news/science-environment-11153466

This item omits to say that it also deflected in off the post, but the deviation from linear is still remarkable. Presumably the paper referred to is in the arXiv, but I’m not going to hunt for that AND the cricket thing.

Anton

5. Anton Garrett Says:

Francis Thompson, the Victorian poet who wrote cricket’s best known poem, At Lords (with the famous elegaic line “O my Hornby and my Barlow long ago”), actually wrote a lighter-hearted poem on cricket after the manner of the Rubaiyat of Omar Khayyam. From this post it’s not hard to see which verse of the original gave him the idea…

6. telescoper Says:

Actually, looking at the footage again I wonder if the change of direction is as large as I thought. The foreshortening effect of the camera makes the angle look large, but taking account of the length of the pitch it may be 20 degrees or even less. That will also reduce the spin speed needed by a factor of a few.

7. Dave Carter Says:

The commentator says it turned two and a half feet, which looks reasonable, but given that the distance from the stumps to the popping crease is three feet, I would think its likely that the distance between the pitch of the ball and the stumps is at least 10 feet. Certainly its too far for the batsman to get a stride forward to smother the spin (which would be about 4 feet, plus the three to the stumps. If it is 10 feet the angle would be 14 degrees.

As Anton says the thing about Warne is his accuracy, together with disguised variation of the direction of the spin. Other bowlers spin it as much. Warne in this instance was bowling into the rough area of the pitched caused by the footmarks of the fast bowlers at the other end, which increases the purchase on pitching, i.e. the efficiency of conversion of rotation into sideways kinetic energy would be particularly high.

Given that I think the angle is about a quarter of a radian, I would think 25-30 revs is reasonable.

Swing bowlers largely get their swing from differences in conditions on the two sides of the ball, not from rotation (the Magnus effect), but I think there may be exceptions (maybe the fast, inswinging yorker sometimes bowled by Shaun Tait is an example of Magnus effect swing). You can also see Magnus effect swing or drift in the video of Warne which Peter posted, if you look carefully the ball swings or drifts away slightly from the batsman through the air, before it pitches and turns back in.

8. Anton Garrett Says:

Mike Gatting: you have been Warned.

9. Dave,

I agree. I think my estimate is too high because I drastically overestimated the angle. I think around 30 rev/s is probably more like it.

I mentioned the Magnus effect in the post, and also pointed out that you can see it in action in the clip. The loop of the ball from the bowler’s hand suggests an initial arc into the batsman, but in fact it clearly swerves away in the air. The classic legspinners line of attack to a right-hander is to to drift the ball into the batsman’s legs, drawing him into an attacking shot and then spin the ball away.

I used to (try to) bowl legspinners in my younger days. I found I could turn the ball quite a lot and could also bowl an authentic wrong ‘un (i.e. googly) as well as a flipper (which is essentially a backspinner). The problem was I sprayed them around all over the place. And it doesn’t matter how much you spin it if it’s a full toss or a wide. I should also point out that it’s actually quite tiring. I found my arm was practically falling off after about 4 overs.

Bowlers don’t usually spin the ball with the spin vector exactly along the direction of motion so there’s usually a combination of off/legspin and topspin/backspin. This poses another problem for the batsman, which is judging the pitch of the ball: a topspun delivery tends to dip sharply and kick on off the pitch. I don’t know if this is empirically verifiable, but I think legspinners tend to generate more variations in bounce than finger-spinners.

Warne used to bowl more googlies earlier in his career, but it puts a lot of strain on wrist and arm. Later on he more-or-less ditched the wrong ‘un and bowled a mixture of legspinners and flippers with the occasional topspinner. If you can turn a legbreak as much as he could then a disguised ball that goes straight on can be just as effective as one that turns the other way.

Peter

10. One thing I should have known as an astronomer is that it’s often very difficult to estimate angles by eye! I should have thought more carefully about the geometry of the pitch. I’ve edited the text to take account of the revisions discussed above. Thanks for the comments.

11. Dave Carter Says:

Peter,

Apologies, I should have read through to the end of the post, and realised that you had already pointed out that you can see Magnus effect drift in the clip. Sorry about that.

The late Ray Lyttleton talked often on this subject, and even gave a talk on the radio. According to his obituary he had wind tunnel experiments carried out to verify his ideas. he was a very keen cricketer, and corresponded with Bradman:

http://www.independent.co.uk/news/people/obituary-professor-raymond-lyttleton-1587695.html

I must admit I never managed to bowl a flipper. I did try legspin, but reverted to dibbly-dobblies when I did play, for much the same reasons of accuracy.

12. Another thing worth mentioning is that Shane Warne’s stock ball was about 60 mph rather than 50 (as I assumed here). That’s not all that slow, in fact. A pub cricketer would find it quite a sharp speed. That makes Warne’s brilliance all the more remarkable.

13. Dave Carter Says:

I remember once in the Parks in Oxford watching the Australian off-spinner Ashley Mallett in the nets at close quarters, and realising that he would be too quick for me irrespective of the spin. Swann would probably be the same, and certainly Shahid Afridi.

• Swann’s a bit slower, I think, and holds it back even more in limited overs games. Afridi is definitely a brisk medium pace.

14. […] Entanglement and Quantum Weirdness After writing a post about spinning cricket balls a while ago I thought it might be fun to post something about the role of spin in quantum […]