However, an undrilled ball with say, an Rg value of 2.28 will react quite differently than a ball with an undrilled Rg value of 2.58, all other things being the same.
First off, an undrilled ball with say, an Rg value of 2.28 would be considered illegal, so I guess it would perform similar to one of those illegal Motiv balls. i.e. Sit nice and pretty on the shelf.
ICDeadmoney- You are correct when you state that if you increase a core's density its moment of inertia (I) increases-- I=mr^2 where m= mass of the imbalance & r= its distance from the center of rotation.
In addition, Rg= the square root of I/M . (M= total mass of the object) , so this shows that increasing a core's density ( mass is the more correct value to use but since mass is directly proportional to density, but it's acceptable to use "density" (D) as long as the object's volume doesn't change -- (D=m/v.) will also increase the Rg value since increasing the core's mass will also result in an increase in the overall mass of the object in question (in our case a bowling ball).
Increasing the density of the core doesn't increase the Rg value.
Lets use a simple bicycle wheel analogy to make the math very simple.
We design a hub shape that has an RG of 1".
We define a rim shape that has an RG of 10".
When we manufacture the hub, and rim out of aluminum, both pieces have a mass of 1 unit. (not sure how big that unit is in reality, but it will be constant so we don't need to know exactly).
We also manufacture the hub and rim out of gold.. In this case both pieces have a mass of 7 units.
Gold is approximately 7 times as dense as aluminum.
Lets assume we make the spokes out of carbon nano tubes so they represent negligible mass... again to make the math easier to follow.
When we use an aluminum hub, and an aluminum rim the math works as follows.
I = 1u * 1" + 1u * 10"^2 = 101
M = 1u + 1u = 2u
Rg = sqrt(101/2) = 7.11"
When we swap the aluminum hub for the gold hub (i.e. increase the density of the core)
I = 7u * 1" + 1u * 10"^2 = 107
M = 7u + 1u = 8u
Rg = sqrt(107/8) = 3.66"
As you see, the moment of inertia increased, while the Rg decreased.
Because moment of inertia and Rg are not directly proportional, it's bad science to say what Nick wrote in the pdf file.
"Thus, in simple terms, the radius of gyration determines how easy it is for the bowling ball of particular weight to rotate about a given axis"
The part that is missing, is as you increase the density of the core, you need to decrease the density elsewhere to maintain the same overall mass.
If we start with an aluminum hub, and a gold rim.
I = 1u * 1" + 7u * 10"^2 = 701
M = 1u + 7u = 8u
Rg = sqrt(701 /
= 9.36"
Increase the hub to gold, while decreasing the rim to aluminum we get:
I = 7u * 1" + 1u * 10"^2 = 107
M = 7u + 1u = 8u
Rg = sqrt(107/8) = 3.66"
In this case we get the expected decrease in moment of inertia as well as decrease in Rg.
One quick thing to note about Rg.
It's about the design of the pieces, not what they are made out of.
It's not until you combine pieces of different density that RG changes to values other than the original design.
Let compare aluminum hub, aluminum rim to gold hub, gold rim.
aluminum
I = 1u * 1" + 1u * 10"^2 = 101
M = 1u + 1u = 2u
Rg = sqrt(101/2) = 7.11"
Gold
I = 7u * 1" + 7u * 10"^2 = 707
M = 7u + 7u = 14u
Rg = sqrt(707/14) = 7.11"
Same Rg, but significantly different moment of inertia.
The overall mass of an object has a lot to do with how easy it will "rev up"
Where torque come in is the friction between the ball and the lane.
In bowling terms, if you want to increase the rate the ball will rev up, you can sand the surface to increase the torque, or select a ball of the same weight with a lower Rg.
If you select a ball with a higher weight but lower Rg, you may end up with a ball that revs up slower.
You can weigh a ball to find it's mass, but not many have an Rg swing to know what the RG is of the PAP.
So you have to get out there and throw the ball for yourself and determine if the results match what you expect when selecting based off the published numbers.
Nick shouldn't say things like increasing the core density makes the ball rev up quicker, decreasing the core density makes the ball rev up slower, and the Rg determines how easily the ball will rev up.
Next thing you know, someone is going to start quoting that bad science as gospel.