## Spinning Out

Posted in Cricket, The Universe and Stuff with tags , , , , , , , , , , on September 6, 2010 by telescoper

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]