Ken

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 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!.

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

Many Thanks

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

Ken Humphreys