# About the Bias

January 11, 2009 by lawn bowls

One of the main things that people will look for when buying lawn bowls is the bias that the bowl has. The bias is the part of the bowl that has been designed to allow the bowl to curve when it is thrown. The bias that the bowl will have will depend on the type of bowl and the manufacturer. Bowls can either have a large bias or a narrow bias and bowlers will choose one based on how they like to throw the bowl. A narrow bias will allow for bowls to get into small spaces and will roll straighter. Large biases are able to go around other bowls and will have a very large curve when they are thrown. How fast the greens are will also depend on the bias of the bowls. There are strict regulations pertaining to biases however, players still have a wide range of bowls with different biases to choose from.

Re. “About the Bias” Can anuone tell me what are the rules that Bowls Manufacturers have to comply with so their woods qualify for the “Stamp” It seems to me that more and more “finer line ” bowls are being advertised and before long we will have some enterprising manufacturer selling bias less bowls.

Your comments will be much appreciated and will go all the way to resolving many a discussion. Thanks.

Ken Humphreys

Hi Ken,

Understand there is a master ‘bowl’ that has the minimum Bias allowed that each menufacture has comply with!!

Thr restamping every 10 years so Bowls can compete in major competitions is to ensue that the set of 4 bowls still are within tolerance of each other and still comply. More importantly it ensures that there has been no tampering with the bowls ie; shaving the bias off to have straighter lines then bendy woods as they progress. (first wood straight, 2nd not so straight etc)

Hope this helps

Hi Tony, Many thanks for your response. Close to resolving the problem except that a little more meat is required onto your first statement that there is a master ‘bowl that has a minimum ‘bias’ In other words who has set the standard and is this standard incumbent on all the manufacturers and all the various National and International bodies. My thanks in anticipation of the problem being resolved.

Ken

If You Want to BOAR Bowlers in a quiz, TRY THIS, looking forward to see who holds the master Bowl; anyway see below;

The trajectory of a ball in lawn bowls

Rod Cross

Physics Department, University of Sydney, Sydney 2006, Australia

~Received 6 November 1997; accepted 26 January 1998!

The main objective in lawn bowls is to bowl a ball along a curved trajectory on a smooth grass

surface so that it stops closer to the jack ~a small white ball! than the opponent’s ball. The curvature

of the path is induced by shaping the ball so that one side is heavier than the other. Some of the

properties of the trajectory are described to illustrate this interesting example of precessional

motion. © 1998 American Association of Physics Teachers.

I. INTRODUCTION

Lawn bowls is a popular outdoor sport for both men and

women, especially with more senior citizens, since the game

does not require a particularly high level of physical fitness.

The game is played at a leisurely pace on an immaculately

manicured lawn and requires only an empirical knowledge of

the laws of precession. Because the ball is weighted on one

side, it travels in a curved path whose radius of curvature

decreases with time until the ball eventually stops. The ball

has a mass typically of about 1.5 kg, and a diameter of about

12 cm, with small variations allowed to suit the individual

bowler and the playing surface conditions. The ball is

launched with a speed of about 4 ms21, at an angle of about

10° to the direct line of sight to the desired end point, then

rolls at a walking pace for about 14 s over a total distance of

about 25 m.

A weighted ball precesses like a gyroscope or a spinning

top1,2 but its axis of rotation is not anchored to a fixed point.

As a result, the precessional motion is combined with linear

motion to generate a curved trajectory. In this respect, the

path of the ball is similar to that of a wheel or a coin3,4 that

735 Am. J. Phys. 66 ~8!, August 1998 © 1998 American Association of Physics Teachers 735

rolls along a surface of its own accord. As the ball or the

wheel slows down, the rate of precession increases and the

radius of curvature decreases. A spinning top, or a rolling

wheel or a coin precesses as a result of the torque associated

with a tilt away from the vertical. A ball in lawn bowls tends

to remain upright throughout its motion since the center of

mass is shifted only slightly from the geometric center and

since the surface is indented slightly by the ball, and the

tendency of the ball to fall over as it comes to rest is resisted

by a sideways reaction force from the ground. The ball will

fall over when it is placed on a horizontal solid surface, and

may also fall over at the end of its trajectory on a smooth

grass surface, depending on its profile and the condition of

the surface.

The ball used in lawn bowls is not perfectly spherical and

it is not deliberately weighted by any additional mass. The

weighting or ‘‘bias’’ is achieved by removing mass so that

the shape remains circular in a cross section normal to the

axis of rotation @as shown in Fig. 1~b!# and is slightly elliptical

in a cross section that includes the axis of rotation @as

shown in Fig. 1~a!#. In the elliptical cross section, the minor

radius on one side of the ball is slightly larger than the other,

with the result that one side of the ball is heavier than the

other. The essential physics can be adequately modeled, both

theoretically and experimentally, using a solid spherical ball

that is weighted by grinding a flat surface on one side. Such

an experiment was described in this Journal by Guest5 in

1965 as an interesting undergraduate experiment in precession.

Guest used a small steel ball bearing on a horizontal

glass or rubber surface and photographed the trajectory with

the aid of a strobe light to measure the linear and angular

velocities. However, he did not derive a formula for the precessional

velocity and did not relate the experiment to lawn

bowls. A simpler laboratory technique is to roll a ball or disk

on carbon paper over a sheet of white paper to leave an

imprint of the trajectory on the sheet.

II. EQUATIONS OF MOTION

The situation modeled in this paper is shown in Figs. 1

and 2. A spherical ball of mass M and radius R is weighted

by removing mass from one or both sides so that the center

of mass is shifted a distance d from the geometric center, as

shown in Fig. 1~a!. The ball is launched in the x direction on

a horizontal surface, in the x – y plane, with an initial velocity

v0 , and subsequently travels in a curved path from O to

K as shown in Fig. 2. The main object of the following

calculations is to determine the correct initial velocity and

direction of the ball so that it finally stops at point K, a

distance D from O and located at an angle D to the x axis.

The problem has previously been examined by Brearley and

Bolt6 and by Brearley,7 but their advanced mathematical

treatment obscures the essential physics of the problem and

is not suitable for any elementary presentation.

The linear motion of a rolling ball can be described by the

relation

v5v02mgt, ~1!

where v is velocity of the center of mass of the ball, m is

the coefficient of rolling friction, g is the acceleration due to

gravity, and t is the time. This relation follows from the fact

that the frictional force acting to decelerate the ball can be

expressed as F5mMg. Experimentally, it is found that

m;0.032 on most bowling greens, independent of the mass

or speed of the ball, but it can be as large as 0.038 on a

slow green or as small as 0.025 on a fast green.6 In the

following treatment, we ignore the initial sliding component

of the motion since the ball starts rolling almost immediately

after it is launched. An unbiased ball will move in a straight

line path of length S5v0 2 /(2mg) and come to rest at time

T5v0 /(mg). The coefficient of friction can be measured

easily, without having to measure the initial velocity, from

the relation S5mgT2/2. The same relation holds for the

curved trajectory of a weighted ball, as shown below. It is

interesting to note that the ball takes longer to arrive at its

Fig. 1. Cross sections of a lawn bowl in ~a! the y – z plane and ~b! the x – z

plane. The z axis is vertical, and the ball rolls in the horizontal x – y plane,

initially in the x direction. For a rolling ball that indents the surface, the

ground reaction force, N, acts as shown in ~b! to decrease both v and v so

that v5Rv at all times.

Fig. 2. Trajectory of a ball from O to K in the horizontal x – y plane. The

ball is launched along the x axis at angle n to OK.

736 Am. J. Phys., Vol. 66, No. 8, August 1998 Rod Cross 736

destination on a fast green than it does on a slow green, since

the ball must be launched at lower speed.

The coefficient of rolling friction8–10 is not simply related

to any other coefficient of friction, for the following reason.

If a ball of radius R starts to slide without rolling, a friction

force, F, acts horizontally at the point of contact, decreasing

the linear velocity, v, of the center of mass and increasing

the angular velocity, v, via the torque applied to the surface.

Rolling commences when v5Rv, at which point there is no

relative motion at the point of contact with the ground. It is

observed that a rolling ball will eventually come to rest. One

might assume that this is simply due to a friction force acting

horizontally on the ball. While such a force would act to

decrease the velocity, v, it would increase the angular velocity,

v, and the ball would end up spinning on the spot like

the wheel of a car stuck in the mud. Rolling friction arises

from the fact that the ball or the surface is slightly rough or

deforms in such a way that the reaction force, N, from the

ground does not act at a point below the center of rotation,

but it acts forward of the center as shown in Fig. 1~b! to

reduce both v and v while maintaining the rolling condition

v5Rv. Since the deformation and hence the reaction force

is approximately proportional to the weight of the ball, m in

Eq. ~1! is essentially independent of M.

In order for a weighted ball to roll smoothly along a

curved path, the axis of rotation must pass through the center

of mass ~CM!, as shown in Fig. 1~a!. If the axis of rotation

does not pass through the CM, then the CM will rise and fall,

generating a wobble in the motion. For simplicity, it is assumed

that the axis of rotation remains horizontal throughout

the trajectory. In fact, the ball will tilt slightly but this has a

negligible effect on the trajectory since the ball tilts only

near the end of its trajectory and usually only by a small

amount.

A frictional force Fy5Mv2/r, shown in Fig. 1~a!, is necessary

if the ball is to follow a curved path. In the absence of

this component, the ball would travel in a straight line, rotating

freely about a vertical axis through P due to precession.

Such a result might be expected, for example, on a

slippery ice surface. The torque component, t x5FyR

2Mgd, acting about the geometric center acts to change the

direction but not the magnitude of the angular momentum of

the ball. This point is discussed in some detail in most elementary

physics texts, in connection with the precession of

a gyroscope or spinning top. The angular momentum is

given by L5ICMv where ICM is the moment of inertia about

the rotation axis passing through the center of mass. At any

point along the trajectory, a tangent to the path makes an

angle f with the x axis, and the change in angular momentum

in the x direction, as a result of a small rotation df in

time dt is dLx52Ldf. The positive x direction is into the

page in Fig. 1~a!. The x component of the torque is therefore

given by

Mv2R

r

2Mgd5L

df

dt

52ICMvvp , ~2!

where vp5df/dt is the angular velocity of precession.

Since v5rvp5Rv when the ball is rolling, Eq. ~2! yields

vp5df/dt5MgdR/~I0v!, ~3!

where I05ICM1MR2 is the moment of inertia about a horizontal

axis through an edge of the ball. This relation differs

from the usual expression for the precessional velocity of a

gyroscope or spinning top in that the relevant moment of

inertia is I0 rather than ICM . Equation ~3! accounts, within

1%, for the experimental data given by Guest.5 The radius of

curvature is given by

r5

v

vp

5

v2I0

MgdR

, ~4!

so

dr

dt

52pr

df

dt

~5!

where dv/dt52mg and p52I0m/(MdR). Equation ~5! can

be integrated to give

r5r0e2pf, ~6!

where r05pv0 2 /(2mg) is the initial radius of curvature. The

path length of the ball is therefore

s5E0

f

r df5~12e2pf!r0 /p. ~7!

The variation of f with time can be obtained by integrating

Eq. ~3!, using Eq. ~1!, to give

f5~ 2/p!ln~v0 /v!, ~8!

indicating that f!` as v!0. From Eq. ~7!, the total path

length is S5r0 /p5v0 2 /(2mg) which is independent of p and

is therefore the same as that for an unbiased ball. The detailed

behavior of the ball at the very end of its trajectory is

not considered in this paper. Equations ~3! and ~4! indicate

that vp!` and r!0 as v!0, but the ball is likely to tilt or

topple in the last few mm of the trajectory.

III. THE TRAJECTORY

The x and y coordinates of the trajectory can be obtained,

as a function of time, from the relations dx/dt5v cos f and

dy/dt5v sin f, so

x1iy5E0

t

veif dt5E0

f

reif df, ~9!

where vdt5ds5rdf. Equating real and imaginary parts of

Eq. ~9! gives

x5

r0

11p2 ~p2pl cos f1l sin f ! ~10!

and

y5

r0

11p2 ~12l cos f2pl sin f !, ~11!

where l5e2pf. The end point of the trajectory, where l50,

therefore has coordinates Y5r0 /(11p2) and X5pY, so the

range of the ball is given by D5(X21Y2)1/25r0 /

(11p2)1/2 and the angle D in Fig. 2 is given by tan D51/p.

A measurement of p can therefore be obtained by measuring

D or X/Y, and the path length can be calculated from the

expression S5(111/p2)1/2D. Since D is independent of v0 ,

a ball must be launched at the same angle to the line of sight,

regardless of the desired range, D. However, it is against the

rules and spirit of the game to use a protractor.

737 Am. J. Phys., Vol. 66, No. 8, August 1998 Rod Cross 737

The radial distance, r, from any point (x,y) to the end

point (X,Y) is given by r5De2pf, indicating that the trajectory

is a logarithmic spiral, as noted by Guest.5 The ball

reaches its maximum displacement in the x direction and

moves purely in the y direction when f5p/2, at which point

v5v0 exp@2pp/4#. The nature of the trajectory is determined

primarily by the parameter, p. If ICM is taken to be

essentially that of a sphere, i.e., ICM50.4MR2, then

p52.8mR/d, which, for d;1 mm, is typically about 5 in

lawn bowls, in which case r/r0;0.0004 and v/v0;0.02

when f5p/2. Consequently, the ball approaches the end

point in a relatively gentle arc and does not spiral inwards by

orbiting the end point more than once, as it would if p were

less than about 1. Typical trajectories are shown in Fig. 3 for

cases where p54 to p510, which are representative of normal

playing conditions. The rules of lawn bowls place no

specific limit on p, since it depends on m, but balls are tested

periodically against a master bowl. To pass the test, the bias

of the ball must not be less than that of the master bowl. One

of the challenges in lawn bowls is to allow for the fact that p

can change during the course of a game since the grass can

dry out during the afternoon and change the coefficient of

rolling friction.

The results shown in Fig. 3 were obtained by rotating

the coordinate system through an angle D so that all trajectories

start and end at the same points ~O and K, respectively!.

In this rotated x8– y8 coordinate system, where the

x8 axis is along the line OK in Fig. 2, one can show that

the maximum y8 displacement is given by y8max

5Dl/(11p2)1/2 and it is located at x8max5D(12py8max),

where l5exp(2pD). For example, if p55, y8max50.0731D

at x8max50.634D and if p58, y8max50.0459D at x8max

50.633D.

Almost identical results are obtained if the ball is allowed

to tilt during the motion. The tilt angle, u, tends to remain

small for lawn bowls, but the trajectory when u varies with

time can be obtained by numerical integration of the equations

dx/dt5v cos f, dy/dt5v sin f and df/dt5A/v

where v is given by Eq. ~1! and A5MgdR cos u/I0 . The

result of such a calculation is that vp differs from the tilt-free

case only near the end of the trajectory, but the trajectory

itself is almost indistinguishable from the tilt-free case. Discussions

of the trajectory of a ball in ten-pin bowling, which

slides rather than rolls for most of the trajectory, can be

found in Refs. 11 and 12.

From a teaching point of view, a rolling ball or coin is

perhaps easier to understand than a precessing top or gyroscope.

A gyroscope or top appears to have the somewhat

mysterious property that it does not fall in response to the

tilting torque, but moves in a direction normal to the applied

force.1,2,13 The same situation applies to a rolling ball or

coin, but a student is more likely to be at ease with the fact

that the ball or coin does not fall, because its momentum

carries it forward.

1P. C. Eastman, ‘‘Painless precession,’’ Am. J. Phys. 43, 365–366 ~1975!.

2P. L. Edwards, ‘‘A physical explanation of the gyroscope effect,’’ Am. J.

Phys. 45, 1194–1195 ~1977!.

3M. G. Olsson, ‘‘Coin spinning on a table,’’ Am. J. Phys. 40, 1543–1545

~1972!.

4R. F. Deimel, Mechanics of the Gyroscope ~Dover, New York, 1952!, 2nd

ed., pp. 93–98.

5P. G. Guest, ‘‘Rolling precession,’’ Am. J. Phys. 33, 446–448 ~1965!.

6M. N. Brearley and B. A. Bolt, ‘‘The dynamics of a bowl,’’ Q. J. Mech.

Appl. Math. 11, 351–363 ~1958!.

7M. N. Brearley, ‘‘The motion of a biased bowl with perturbing projection

conditions,’’ Proc. Cambridge Philos. Soc. 57, 131–151 ~1961!.

8J. Witters and D. Duymelinck, ‘‘Rolling and sliding resistive forces on

balls moving on a flat surface,’’ Am. J. Phys. 54, 80–83 ~1986!.

9A. Domenech, T. Domenech, and J. Cebrian, ‘‘Introduction to the study of

rolling friction,’’ Am. J. Phys. 55, 231–235 ~1987!.

10J. Hierrezuelo, B. R. Catolicos, and C. Carnero, ‘‘Sliding and rolling: The

physics of a rolling ball,’’ Phys. Educ. 30, 177–182 ~1995!.

11D. C. Hopkins and J. D. Patterson, ‘‘Bowling frames: Paths of a bowling

ball,’’ Am. J. Phys. 45, 263–266 ~1977!.

12R. L. Huston, C. Passerello, J. M. Winget, and J. Sears, ‘‘On the dynamics

of a weighted bowling ball,’’ J. Appl. Mech. 46, 937–943 ~1979!.

13J. B. T. McCaughan, ‘‘Teaching gyroscopic precession at the elementary

level,’’ Phys. Educ. 17, 133–138 ~1982!.

What a post Keith!!!!!

Real answer is that the World Bowls licnce the table and teating equipment to confir with tere ‘MASTER BOWL as below!!!!¬

STAGE 6

Final testing using the approved and licensed test table. All bowls are tested to the W.B.B. “Master” bowl. As well as having to comply with the even more rigorous requirements of the manufacturers. Thus the different models are tested to their model specifications to confirm that their performance will be optimal for given greens.

The response from Keith made me check the date and finding it was not April 1 I can only assume he suffers from both verbal diarrhoea and basic comprehension.

Thank you Tony for a succinct and literate answer.

Ken

As a coach and keen mathematician I have always wanted to know the maths of the curve a bowls takes and so I found this exposition both interesting and bewildering. I wonder do manufacturers use mathematical formulae in order to determine the bias of a new bowl or is it hit and miss.

I’ve ‘read’ Keith’s entry on the motion of a bowl on the trajectory of a ball in lawn bowls by Rod Cross, Physics Department, University of Sydney, Sydney 2006, Australia.

I also have an interest in this area (please don’t tell me ‘to get a life’), I do have one.

Does anyone have access to the original article as the maths contained in it is somewhat garbled.

Hi, basic question from a newbie for all you bowlers.

Given green conditions, wind, temperature and bowl bias how may paths will take a given bowl to the jack (or any desired location ?) Is it only one or range?

Suppose you find the combination of speed and angle that gives you the perfect path. Now , if you bowl a narrower angle than that of the perfect path, the bowl has to have less speed (because it is travelling a shorter distance) . Will it , however, curve off the line or can it curve less? Suppose on the other hand you bowl at a wider angle than the perfect path. In this case, you have to have more speed because it needs to travel further. Can it curve back enough or will it stay wide?

I’ve had a look at the above article by Cross. The answer may be in there somewhere, but I can’t find it. In any case, anyone who has bowled for a while (and has not been kidding themselves! ) will know the answer.

Keith’s response is a straight copy of a theoretical academic paper by Rod Cross published in August 1998 in the American Journal of Physics, for Physics Teachers.

The rules and regulations for master bowls are determined by Word Bowls Ltd. For more information you can visit their website.

In answer to Rob Buttrose’s query about bowl delivery direction, I have looked at Rod Cross’s piece and had e-mail ‘discussions’ with him.

What emerges from his work, amongst other things, is that to bowl to a given jack position, the angle of delivery (that’s the angle between the line between the mat and jack (centre line) and the initial direction of motion of the bowl) is constant regardless of the jack length.

So if you bowl narrow with reduced weight, the bowl will cross the centre line and end up short, and vice versa.

Arguments often arise about the statement ‘you need to give the bowl more green for a longer jack length.’ What Rod Cross’s maths shows is that for a constant mat position, the angle of delivery, therefore the ‘aimimg-point’ on the rear bank, is constant. True for a longer jack length the bowl will go further from the centre line.

However, with a varying mat position and fixed jack length, again the delivery angle is constant, but the required aiming point moves inwards towards the centre line as the mat is moved up the green. This confuses some bowlers.

I’m happy to explain this further if anyone is interested.

How do you decide what weight,size and bias suits you?

I started with size 3 and wide bias and my son has bought me size 3 with a narrow bias.Both of which I was ok with but not consistent.

I see other players with various sizes with heavy added,how do you know which is right for you to get consistency.

The important points to consider when choosing a bowl type (narrow or wide) assuming you have the correct size and weight are:-

The amount of bias a bowl has and the type of curve it takes.

The speed/s of greens you normally encounter i.e. grass, synthetic and carpet.

The position you normally play in a rink.

Remember when you use narrow bias bowls you have less tolerance with your weight (length of travel) than with a wider drawing bowl.

With the narrow drawing bowl you have more tolerance with your green (width) than with a wider drawing bowl.

On the slower greens you can get away with narrow bowls if you play at the ‘front end’, if your’e a ‘back end’ player you might be better off with a wider drawing bowl.

There is a strong body of opinion that some modern narrow bowls are more affected by the wind than their wider counterparts.

On the faster greens (15+ seconds) the ‘back end’ players prefer a narrow bowl because it has less or no hook at the end of it’s travel. This makes the ‘yard-on’ shot and it’s over draw weight deratives much easier to play.

Hi,

Just for an experiment I am interested in instrumenting a lawn balling bowl with a 9 DOF IMU package. This would allow me to track and display the trajectory, speed, direction, ‘wobble’ and path of a ball in real time on a computer screen. The IMU has three 3 axis sensors, an accelerometer, a gyro, and a magnetometer. The gyro maybe limited to 2000 degrees per second , which tranlates into about 5.5 revolutions per second. Can anyone tell me the range of ball release velocities when balling. (I may be limited to about 9.8 mph using a 5″ diameter ball?)

One could also use it to measure the ’speed’ of the green, the profile of the green observe the affect of bias etc.

regards

I have just started playing bowls and am having difficulty deciding how to face my ball depending on the direction. Which side depicts the bias one way or another

Hi,

Where can I purchase a set of the lowest bias bowels?

Thanks

Larry

I have a question.

In a singles game where each bowler delivers four bowls, do all four bowls have to be identical or is it legal to have different biases on each bowl?

As bowls are turned on a lathe there isn’t really any need for such fine calculations as the bias can be made simply by turning the wood or plastic rough blank to pure spherical with a spherical turning attachment – less the very small points needed to provide grip between centres of the lathe – and then both increasing the radius of the attachment to make a slightly bigger sphere while simultaneously moving it a set amount to one side – this will make the cutter remove material from the inner portion of that side without removing any from the diameter so that the second cut half has less material and when rolled will fall away or bias to the other side (I use ‘bias’ as a verb and a noun).

Different biases can be made by different second cuts as the amount of material needing removal will be quite small which is why many people cannot tell a bowl is not a perfect sphere while old precision turners like myself can see that some bowls are definitely not spherical but just where and by how much eludes us.

Before spherical turning attachments were available a bowl could be biased by making a semicircular template and shaping the bowl in a plain lathe – as commonly used to make bulbous table legs etc – and cutting the bowl by hand tools and eye until it fit the template then removing material at one side until a gap showed between template and bowl and that would be the lighter non-bias side and again variations in bias could be made by altering the gap width.

These days a computer controlled lathe is used and the bowl designer can plot a path for the cutter to follow to make a bowl and can make the cutter move in a non-semi-circular path to ensure the outer rolling diameter is perfectly spherical but some part of one side is underspherical or flattish and the opposite side is overspherical or bulging and thus heavier and will be the bias side.

Sets of flat/lawn/carpet bowls are intended to have different biases so that #1 will roll with slight bias, #2 with more, #3 more and #4 lots more so that as the jack becomes blocked by earlier bowls the later ones can be sent around the blockers to hit the jack.

A lignum vitae perfectly spherical bowl may have a natural bias due to the density of the grain due to conditions when the wood grew or how big a tree it was cut from as in many trees the wood density can vary from outside to core.

The variations or lack of variation in lignum bowls may have brought about the use of metal weights fitted to one side to try make pairs of bowls that biased the same.

can someone tell me if there is a differance between the bias of a ball used for crown green and lawn green balls