How do you measure eccentricity?

...if the centre of gravity were to be in the same place for every single ball, then one cannot blame manufacturing difficulties.

Here's a quick and dirty test you might try:

If your interest is whether the balls are the same, then I would think you could place a ball on a hard, level surface and let it come to rest. The CG should be on a vertical line going through the point of contact and closer to the bottom.

Mark the top of the ball and roll it so the marked spot is 90 degrees from the vertical and let it return to equilibrium. Do this with several balls and see if the action is the same.

If the action is slower for a ball, the CG is closer to the center, assuming all the balls have about the same moment of inertia.

I understand what you are saying, but the assumption needs to be that all the balls have the same mass. Varying times to return to balance will be caused by varying moments of inertia, themselves due to varying distances of the CG from the centre. The problem, though, is that the eccentricity is very small in the first place.

Here's another idea on how to locate the CG.

Allow the ball to reach equilibrium on a hard, level surface and mark the topmost point. The CG will be on a vertical line through the point of contact and the topmost point, closer to the bottom. Now, if you roll the ball so that the marked point is level with the center of the ball, there will be a maximum force to return the ball to equilibrium.

This force can be countered by putting the ball on an inclined plane, elevated at an angle alpha, adjusted to balance the force.

The ball's radius is R and the CG is offset by distance r, to be determined. If the ball is rolled up the slope at a tiny angle θ radians, the CG is raised in elevation by R θ sin(α) and lowered, due to rotation, by r θ. If α is adjusted so these are balanced:

r θ = R θ sin(α)

or

r = R sin(α)

you could do this by floating them in a bowl of water. besides spin see how deep they float.

edit:

I see I should have read the rest of the replies before posting.

I would spin it on a perfectly flat surface...if it spins without moving on the table, mark the top and bottom....if you can do this at every point on the ball, then the ball is perfectly balanced...as the ball slows it will drift, make a scattering map of the final location when stopped, that pattern should be randomly distributed...

...or you can try this method...

...."A ball may be perfectly spherical but have an uneven distribution of weight within its bulk. This will cause it to deviate from a straight path like a bowls ball. The centre of mass can be found by floating the ball and seeing if it has a marked preference for settling in one orientation. (See the article by Richard Le Maitre).

One problem is to float the ball. A croquet ball will sink in water, its density should be 1.1089 g.cm^-3, water is 1.00g.cm^-3. You need a denser medium. Calculations indicate that you would need a 15% by weight salt solution. That can be prepared by dissolving ~166g of salt per litre of water at 20ºC (unchecked!). Add more salt to increase the floatation, or water to reduce it. The degree of bias can be measured by attaching a weight to the surface of the ball until the bias is countered. For really sensitive measurements it has been suggested that some surfactant (detergent) should be added to the brine to lower the surface tension. Reports indicate that some balls require as much as a 5g weight. See recent article (2008)."...

http://www.oxfordcroquet.com/tech/testing/index.asp

I thought about putting them in water, but if they are not sealed, absorption of water would complicate the situation...

I'm thinking they would be varnished, but an old set you might be right...of course the modern ones would be plastic...

$175 on Etsy...

$783. on 1stdibs.....

I would spin it on a perfectly flat surface...

If you spin the ball, I believe the heavy end will point up, just the opposite of letting it roll to a rest on a flat, level surface. I think it's a combination of friction with the surface and gyroscopic action. Here is an example using a spinning ring...

Well I think it depends on the speed of the spin...If the ball starts out in one attitude and corrects to another, you might be able to measure the offset...if on the other hand the ball is perfectly balanced and spinning on a flat surface it shouldn't move...a wobbling ball on slow spin is unbalanced...perhaps the unbalanced amount could be measured by the speed of spin necessary to correct balance....

Dump them in a bucket of water. The CG will become evident.

Question is, what will you do then?

That is true only if these balls float in water and if the geometry of the balls are sufficiently close to an ideal sphere. If both conditionals are met then floating orientation stabilities will indicate if the center of mass is close enough to the geometric center.

However, if this is a geometric eccentricity concern then this becomes much more difficult. The "high tech" and most precise approach will be to laser scan the balls in three dimensions. Then in a computer analysis of the digital model take multiple slice measurements of circumferences.

The most practical geometry test for eccentricity would be with a Vernier, dial or digital caliper and measure the diameter of each ball for six to a dozen different orientations. Perfect spheres will present identical diameters regardless of orientation.

If the original manufacturing process tested for geometric eccentricity, they had ideal or standard cups each finished ball had to not excessively wiggle in any orientation.

You still bring up an excellent point. Outside of satisfying the personal curiosity of why some balls will seem to roll predictably in a curve, what will one do with the ball(s) once the cause is found?

Here's my twopence worth.

Place the ball on a flat even surface, and, let it come to rest; mark the "equator"; hold the ball with the equator vertical and find the point where it as nearly as possible balances on the point of a pin.

The distance between the equator and balance point is then the distance of the C of G from the true centre of the ball.

Are you sure you haven't got a box of carpet bowls?

I believe you’re confused with what you’re are really looking for.

you’re not looking for the center of gravity, you looking for the moment of inertia.

CG only equals CM when gravity is uniform across the object. The question is, is it uniform?

“Moment of inertia describes mass and position of that mass with relation to the axis of rotation. How hard it is to spin. A wheel is harder to spin than an ball of the same mass.

Newton's second law describes a phenomenon that we call inertia. This is mass in the equation F=ma. But from a spinning reference point it is moment of inertia.” Moment of inertia describes mass and position of that mass with relation to the axis of rotation.

How hard it is to spin. A wheel is harder to spin than an ball of the same mass. Newton's second law describes a phenomenon that we call inertia. This is mass in the equation F=ma. But from a spinning reference point it is moment of inertia.”

I would suggest mounting the balls between two points, similar to the unit used to set spokes in a bike wheel, with minimal pressure to retain the ball and let it move to the point where the center of gravity pointed down. Do several tests.

The points could be modified with a small ball bearing ball in place of a point. In the early days of power meters the rotary disk had a tip with a ball bearing and spun on a sapphire cup, the top was held by a hardened steel pin in a brass cup.

Just a thought.

Using a pentel pen I would randomly mark a dot each ball and drop it in a pail of water and observe where the pen marked on the ball will settle..

I believe you'll have a pretty good idea how evenly the mass is distributed and where the center of gravity would be..

Thank you all. I particularly appreciated Rixter's suggestion in #3. Looking more closely at the balls, they appear to have been turned between centres, and the lack of sphericity is another reason for the erratic paths (max measured diameter 38.04 mm, min 36.63 mm)

Non-sphericity pretty much makes it impossible to analyze, but it was an interesting problem to play with.

Perhaps they were spherical when made. Wood is an anisotropic material due to the grain direction. It would be interesting if the non-sphericity and grain direction were correlated.

Wood is also nonhomogenous. Solid shapes may have pitch pockets, knots, voids...

OP tells us miniature, this leads me to think toy size. Would a manufacturer spend a lot of effort to ensure kids game gear met the same tolerances as full size?

Just a thought...

If the wood is not sealed, the balls might have been stored in a damp place, for example, a concrete floor of a basement. Absorbed moisture on one side could account for the offset center of gravity.

The using the CG is useless if the object isn’t uniformed., you’ll then need to find the moment of enertia.

“Generally, for uniform objects, the moment of inertia is calculated by taking the square of its distance from the axis of rotation (r2) and the product of its mass. Now, in the case of non-uniform objects, we can calculate the moment of inertia by taking the sum of individual point masses at each different radius.”

The moment of inertia can be measured with a trifilar pendulum. Basically, the object to be measured is suspended by three wires. It is twisted and released and the frequency of oscillation is measured, and from this and the mass, the moment of inertia can be calculated.

Dynamics: Trifilar Pendulum - Paul Titchener

For an object that is not perfectly symmetrical, the moment of inertia will depend on the axis chosen. To compare the croquet balls, the corresponding axes could be "allow the ball to reach equilibrium on a hard, level surface and mark the topmost point."

The balls could have the same moment of inertia and still be different, but it's another measurement.

First you would have to get the ball perfectly round...then mark the entire surface of the ball with a numbered grid...then bounce the ball from about three feet on a perfectly flat and level hard surface, monitoring the ball with high speed camera...

..dropping the ball from each number location on the ball at the top, you could then measure the deflection angle when the ball bounced and map the internal center of gravity by logging the deflection from each location...

..at the point of most deflective bounce would be the location of the furthest point from center in the direction of the bounce, of the center of gravity...

..and it's location could be confirmed by bouncing the ball with the number location from the side now at the top, should bounce straight...

phph001 has now told us that the balls are not spherical, but, whilst we believed that they were, many people advocated putting them in water.

I am intrigued to understand why floating them in water is a better idea than simply allowing them to come to rest on a flat horizontal surface.

What am I missing?

No friction = more accurate

Me too. Marking the top of a floating object is not as easy as marking one that is held down on a hard surface.

With a Rorschach test?