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Dr Ian Plummer

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Technical
Further Lawn Speeds Calculations

Currently there are two units used in lawn speed measurement :

In the former the time taken for the ball to travel a set distance is measured; in the latter a ball is rolled down a standard ramp and the length of its travel measured.

This article extends on the discussion on Lawns Speeds and attempts to show the following:

  1. Conversion between Plummers and Nels
  2. How the Ball Composition affects Rolling Velocity
  3. How the Angle of the Ramp affects the Nel Value
  4. The Effect of Lawn Slope on Measurements

The basic mathematics was worked out by Nick Furse and Samir Patel with corruptions added by the author.

1. Conversion between Plummers and Nels

Measurements at The Hurlingham Club a few years ago suggest that constant deceleration (a) is a good approximation for a rolling croquet ball (see the article by Plummer). So using this assumption:

  v = ua.t, hence u = a.t
(1)
 
  d = u.t – ½.a.t2 = ½.a.t2 = u2/2.a  
(2)

If we further define:

 

D =

distance from North to South boundary = 105 ft (35 yards)

 

P =

(Plummers) time take for a ball to travel from South boundary at stop on North boundary

then we can also determine that the constant deceleration (a) is

 

D = ½.a.P2; or a = 2.D/P2

by substituting D=d, P=t, etc. in eqn (2)

(3)

Then, if we further define

 

=

horizontal velocity of ball on leaving ramp

 

N =

(Nels) the distance a ball rolls from the bottom of the Nel ramp

and we reapply (2) given V, the initial velocity (=u), and substituting for a we can find the distance travelled in Nels

 

N = P2.V2/(4.D)

So, Nels are proportional to the square of Plummers.

 
(4)

The constant of proportionality is V2/(4.D).  Given that D is fixed it is V which must reflect possible differences.  This is investigated below.

2. How the Ball Composition affects Rolling Velocity

In this Section we produce a relation between the rolling and translational energies and hence velocity of the ball at the bottom of the ramp.

The factor being concentrated on is the moment of inertia (moi) of the ball. This arises because some croquet balls are solid and others hollow and this will affect the rolling properties. As an example, a solid sphere of aluminium and a spherical shell of gold, both the same size and mass, will roll down a slope at different rates.

The following analysis assumes that the properties of the ball affect only the Nel measurement and not the Plummer measurement. Whether this is valid still remains to be tested - for example by using the timing measurement (Plummers) with different composition croquet balls on the same lawn.

In Section 3 we will consider what happens when a ball rolling with a translational velocity and angular velocity (rate of revolution) hits the grass at the end of the ramp. 

We apply the principles of conservation of energy, but first define:

 

H =

the height of the back of the Nel ramp

 

U =

velocity of the ball along the ramp at the bottom of the ramp

 

g =

acceleration due to gravity which is approximately 9.8 ms-2 or 32 ft s-2

 

m =

the mass of a croquet ball; 1 lb or 0.454 kg

 

r =

the radius of a croquet ball; 1.81" or 46 mm

 

p =

the density of the material from which the ball is made

 

=

the moment of inertia (moi) of a croquet ball

 

=

the rotational velocity of a croquet ball which is rolling with transitional velocity U, (w = U/r)

So for a ball starting some height, H, above the ground, all the Potential Energy (PE) due to its height will be converted into Kinetic Energy (KE). The Kinetic Energy can be separated into the translational and rolling components:

  PE = KE(translational) + KE(rotational)
conservation of energy
(5)

Substituting for the standard (text book) energy terms yields:

  m.g.H = (½.m.U2) + (½.i.w2) = (½.m.U2) + (½.i.(U2/r2))  
(6)

The moment of inertia for a homogeneous solid ball is well known: i = (2/5).m.r2. What we wish to consider is a ball with the hole in its middle (such as the Barlow balls), where this hole has radius (o.r). Here o is a factor between 0 and 1. The moi of the hollow ball can be calculated by considering a solid sphere's moi then subtracting the moi of a solid sphere equivalent in size to the void.

Define M = m + m1, where M is the sum of m (the hollow shell) and m1 (the mass of the filled void). Thus we have:

  i = (2/5).(m + m1).r2 - (2/5).(m1).(o.r)2 i.e. moi of hollow sphere =
moi of a solid sphere - moi of absent sphere
(7)

We can also use the relationship between mass, volume and density to relate the mass to the radius:

  M = (4/3).PI.r3.p, where p is the density mass = volume.density
(8)

(4/3).PI.r3 is the standard equation for the volume of a sphere, then from (8), m1 = M.o3.

 

m1 = (m + m1).o3 = m.o3 + m1.o3 since M = m + m1
(9)
 then m1.(1 - o3) = m.o3 and m1 =  m.o3/(1 - o3)
(10)

Plugging this in to the term for the moment of inertia above (7) produces:

  i = (2/5).m.r2.[1 + o3.(1 - o3) -o5/(1 - o3)] or
(11)
  i = (2/5).m.r2.[(1 - o5)/(1 - o3)]
(12)

For ease let B, a factor unique to the ball (depending on the size of the hole in the middle and consequently its density), be defined as:

  B = (1 - o5)/(1 - o3)
(13)

Hence

  i = (2/5).m.r2.B from eqn (12)
(14)
  PE = KE
(15)
  m.g.H = ½.m.U2 + ½.i.(U2)/(r2) i.e. eqn (6)
(16)
  m.g.H = ½.m.U2 + (1/5).m.r2.B.(U2)/(r2) substituting for i
(17)
  m.g.H = m.U2.(½ + (1/5).B)
(18)
  10.g.H = U2.(5 + 2.B) by multiplying through by 10
(19)
  U2 = 10.g.H/(5 + 2.B)
(21)
 
U = √[(10.g.H)/(5 + 2.B)]
(22)

We now have an expression for the translational velocity of a ball at the bottom of the ramp in terms of only H and B, which itself is depends on the size of the void in the ball. Note the velocity is down the ramp not along the grass. As a test, if the ball is solid o = 0, from (19), we get m.g.H = 7/10.m.U2 - the standard result as in "Properties of Sliding and Rolling Croquet Balls".

3. How the Angle of the Ramp affects the Nel Value

In the last Section we showed that, dependent on the construction of the ball, there will be different amounts of Kinetic Energy locked into rolling and translation, resulting in different translational velocities at the bottom of the ramp. We now establish how this is converted into the velocity of the ball rolling along the lawn once it has left the ramp.

We assume that, at the transition from the slope to the horizontal, the horizontal component of the translational energy is maintained whilst the vertical is lost (e.g. by damped bouncing against the grass) and the rolling energy is maintained. It is then assumed that the rolling and translational energy are repartitioned without loss to produce a ball which rolls and does not skid.

Again, intuitively, if the slope were at 90º (vertical) and a ball dropped down it you would not expect the ball to go anywhere.  However if a spinning ball was dropped, you would expect it to kick off in the direction of the spin.

U is the velocity of the ball travelling down the slope (i.e. at an angle towards the ground). U has a horizontal (Uh) and vertical (Uv) components of velocity, such that

  Uh2 + Uv2= U2
(23)

Let A be the angle between the ramp and the ground, hence

  Uh = U.Sin(A) and Uv = U.Cos(A)
(24)

When the ball hits the ground at the bottom of the ramp, the rotational velocity of the ball will be greater than the horizontal velocity of the ball. Then it is assumed that the rotating ball will increase its horizontal velocity by conversion of some of the rotational energy into horizontal 'transitional' energy until there is no skidding. For discussion on rolling and sliding see "Properties of Sliding and Rolling Croquet Balls".

When the ball leaves the slope, let V be the new horizontal velocity, which will be the combination of the horizontal velocity due to the slope of the ramp and the spinning of the ball in order to conserve (Kinetic) energy:

  ½.m.V2 + (1/5).m.V2.B = ½.m.Uh2 + (1/5).m.U2.B conservation of energy as eqn (17)
(25)
Since Uh2 = U2.(Cos(A))2, then V2.(½ + (1/5).B) = U2.(½.(Cos(A))2 + (1/5).B)
(26)
or V2 = U2.(5.(Cos(A))2 + 2.B)/(5 + 2.B).
(27)

This gives the real horizontal velocity of a ball after leaving the ramp. Just to test equation 27: if the angle of the ramp is very shallow (Cos(A) ≈1) and the ball is solid (o = 0) then V = U.

We can now return to the realationship between Plummers and Nels and insert the results derived in the last two sections.

We now use the updated value of V in eqn (4), which incorporates the moi and the partitioning of the energy due to the angle of the ramp.

From the Plummer test, we know that the deceleration along the lawn is: a = (2.D)/(P2), then the distance that the ball will travel from the bottom of the ramp when travelling at velocity V is

  N = (P2/(4.D)).V2 from eqn (4)
(28)
  N = (P2/(4.D)).U2.(5.(Cos(A))2 + 2.B)/(5 + 2.B) substitute for V2 from in (27)
(29)
  N = (P2/(4.D)).10.g.H/(5 + 2.B).(5.(Cos(A))2 + 2.B)/(5 + 2.B) substitute for U2 from (21)
(30)

this can be tidied up slightly as:

  N =P2.(5/(2.D)).g.H.(5.Cos(A)2 + 2B) / (5 + 2.B)2 where B = (1 - o5)/(1 - o3) from eqn (13)
(31)

Thus we now have a conversion of Plummers to Nels involving the height and angle of the ramp and the relative moment of inertia of the ball via B. We can test (31) to make see if it gives a one off sensible result. Using

 
D = 
105 feet
 
H = 
1 foot
 
Cos(A) = 
(√15)/4 (If the ramp is 4 foot long)
 
g = 
32 ft.sec-2
 
o = 
½ (an assumption)
 
P = 
12 sec

We get N = 14.5 feet which is close to the experimental results.

4. The Effect of Lawn Slope on Measurements

It has been observed that different timings (Plummers) were measured when shooting along the nap of the lawn or against it. Whilst it is clear that the nap will offer a directional coefficent of friction (think of stroking velvet!), a sloping of the lawn would produce the same effect. The lawns at Parkstone in May yielded 9 Plummers against the nap and 10 with it.

A 10 Plummer lawn implies an average deceleration of (2 x 105/102) = 2.1 ft/s/s, whilst a 9 Plummer lawn implies an average deceleration of 210/92 = 2.6 ft/s/s

The average deceleration is thus 2.35 ft/s/s. If the variation is attributed to a slope in each direction, then the increase/decrease in deceleration of 0.25 ft/s/s would be caused by the gravitational effect of the slope. If the height of the slope is 'h' then: h/105 = 0.25/g where g = 32.2 ft/s/s. Thus h = 105 x 0.25/32.2 = 0.815 ft.

Thus a genuine difference of speed of a lawn in each direction (of 9 and 10 Plummers) is equivalent to a slope of length 105 ft and height 0.815 ft or about 10 inches.


Experimental Results

The following measurements compare near simultaneous measurements of Plummers with Lamberts. This table will be updated as more results become available.

Date Location Lamberts
(feet)
Plummers
(seconds)

15-Apr-06

Surbiton

11.5

10

16-Apr-06

Surbiton

12

10.3

29-Apr-06

Hunstanton

13

11

Comparison of Lamberts versus Plummers

Samir reports: Westerns, May Day, Parkston. After a day's play, it became clear that there was a difference between playing with the 'nap' (i.e. in the direction the mower went) rather than against it. With the nap, the lawn speed was 10 Plummers; against it 9 Plummers.

Author: Dr Ian Plummer
Samir Patel
George Noble
Mike Lambert
Nick Furse
All rights reserved © 2006-2017


Updated 28.i.16
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