After the first day of the third Ashes test cricket match between England and Australia it may be a good time to consider how spin bowling might affect the outcome of the series - and how science can be used to describe the effects of spin bowling.
In our recently-published paper in the journal Physica Scripta we mathematically modelled the effects of spin and wind on a rotating spherical projectile, such as a spinning cricket ball.
The basics of spin bowling
Spin bowlers attempt to deceive batsmen through movement of the ball through the air, and by deviation of the ball as it bounces off the wicket. Ball movement through the air occurs as a result of the angular velocity imparted to the ball by the bowler.
This movement can manifest itself as a “drift” away from the batsman or a drift towards the batsman. The ball can also “hold up” (or “lift”) so that it bounces, or pitches, further down the wicket towards the batsman than if the ball had no spin. Alternatively, it can “dip,” so that it pitches shorter than it would had it had no spin – as English test cricketer Graeme Swann does to great effect in the video below:
There can also be combinations of drift and lift, or drift and dip.
A right-handed finger spinner will predominantly bowl off-spinners, in which the ball rotates approximately clockwise (from the bowler’s perspective) as it travels down the pitch, and typically turns off the pitch from the off-side (the right side of the wicket for a right handed batsman) towards the leg-side (the left side of the wicket for a right-handed batsman).
A right-handed wrist spinner will predominantly bowl leg-spinners with the ball rotating approximately counter-clockwise (from the bowler’s perspective) and typically turning off the pitch from the leg-side towards the off-side.
Of course, this is an over-simplification because class bowlers of both varieties will have quite a few variations of deliveries that they can bowl to produce different movements through the air and off the pitch. Swann talks through some variations in the video below:
They will also attempt to disguise their bowling action in order to convince the batsman the ball they’re receiving is something other than what it really is.
Use the force
A spinning ball has three forces acting on it:
- a gravitational force acting vertically downwards
- a drag force
- a lift or “Magnus” force, which occurs only if the projectile is spinning, as shown in the diagram below:
Cases in which the spin vector is perpendicular to the projectile velocity (“top-spin”, “under-spin” or “side-spin”) have been studied before but cases in which the spin vector is in an arbitrary direction and in the presence of a wind are significantly more difficult to treat.
Such arbitrary spin directions are of interest for a spin bowler’s delivery in cricket, where the spin axis may be approximately in the direction of the velocity vector, at least in the initial stages of the flight.
In our calculations, we made some assumptions about the motion of a cricket ball, including:
- The ball is spherical
- The effects of the seam are ignored
- The drag force is proportional to the square of the velocity and acts in the opposite direction to it
- The lift force is proportional to the square of the ball’s velocity and acts perpendicular to this velocity and to the angular velocity vector
The diagram below shows the trajectory of an off-spin delivery, or “wrong ‘un” as delivered by a leg-spin bowler. Such a delivery may exhibit sideways drift in flight and, in the presence of a crosswind, may dip or lift vertically before pitching.
In our study, we quantified the effects of a cricket ball’s spin, in the presence of crosswinds and headwinds, and showed that the ball can exhibit sideways drift and vertical dip and lift.
To demonstrate the effect of spin and wind on the sideways motion of a cricket ball as it travels through the air before pitching, consider the trajectories in the diagram below of five deliveries bowled from a height of 2m, at 79km/h and an angle of elevation of 5°, with:
- No spin (solid blue line)
- Pure off-spin of 1,432rpm, no wind (purple dashed line)
- Pure side-spin of 1,432rpm, no wind (black dashed line)
- Off-spin of 1,432rpm into a headwind of 27km/h (light blue dotted line)
- No spin, but with a 5.4km/h crosswind from the leg side (solid red line)
The two off-spin deliveries first drift slightly to the leg-side as the ball rises and then to the off side as the ball falls, the effect at 18m down the pitch being more noticeable if the headwind is present (2cm) than if there is no headwind (1cm).
For pure side-spin the effect is far more dramatic, even with no wind, the ball drifting to the off side by 10cm after having travelled only 11m down the pitch. It would therefore appear that the bowler can control the amount of drift by altering the mix of side-spin and off-spin components – something which is, no doubt, well known to spin bowlers.
Finally a delivery with no spin in the presence of a sidewind is displayed where a sideways movement of about 8cm results by the time of pitching.
To see the effect of crosswinds, we calculated the trajectories of three off-spin deliveries, rotating at 1,432rpm, and with the same height, speed and angle of elevation as the examples above, in the presence of three wind conditions:
- No wind (solid blue line)
- A crosswind of around 14km/h blowing across the pitch from the off side (black dashed line)
- A crosswind of around 14km/h blowing across the pitch from the leg side (red dashed line)
There will be a small sideways drift of about 1cm to the off side in the no-wind case. This drift is purely due to the Magnus force as a result of the rotation of the ball.
The presence of a cross-wind completely dominates this effect, as is expected, and moves the ball in the direction of the wind by more than 20cm, the effect being larger when the wind aids the drift due to spin, again as expected.
Vertical dip and lift
If we look at those same three deliveries from the side, we can see how crosswinds, combined with spin, can affect the length at which a delivery pitches.
The right hand graph below also shows a delivery with no spin in the presence of a 14km/h crosswind from the leg side (dotted purple line) for comparison.
A wind blowing from the off side causes the spinning ball to lift above the no-spin trajectory, and a wind blowing from the leg side causes the ball to dip below the no-spin trajectory.
The variation in height from the no-spin trajectory is about ±4cm and, more importantly, the variation in the point of pitching is about ±14cm.
Even a more modest wind of about 1m/s (3.6km/h) would cause a variation in the point of pitching of about ±3.5cm.
The trajectories for the no-wind case (solid blue line) and the no-spin case in the presence of a crosswind from the leg side (dotted purple line) in the graph on the right lie very close together and approximately central between the other two lines.
Importantly, this indicates that it is the combination of the crosswind and the spin - not the spin or wind alone - which causes these effects.
These sideways and vertical movements can be large enough to cause the batsman to either misjudge the position of the ball and “edge” it to a fieldsman, or to misjudge the point of pitching and lift it in the air.