Squirt. End Mass and Cue Flexibility.

[LAMas]

What Rod Cross was studying in the OP.

VII Conclusion

....The experimental data shows that elasticity of the cue tip plays a dominant role in the collision process and suggests that cues with thin shafts might generate lower squirt angles as a result of their greater flexibility rather than their lower mass...


http://www.physics.usyd.edu.au/~cros.... squirt.pdf

Dave's link:
Without going into details, it is worth noting that the interaction of the cue and the cue ball is not limited to their deformation only at the point of contact, but is also related to the distribution of longitudinal waves of compression and relief in them. It is the transmission of these waves that defines the time of contact between the cue and the cue ball as well as the fraction of energy transferred from the cue to the cue ball. In addition, the longitudinal load of the cue contributes to its loss of stability and lateral deformation and, consequently, to the generation of transverse oscillations which also absorb energy transmitted to the cue ball. We should not forget that any lateral deformation of the cue during the stroke can lead to significant deviations of the cue ball from its intended trajectory. All these factors suggest that in describing the mechanics of the cue, we need to considered its wave properties in addition to its geometry and weight characteristics.

http://dbkcues.ru/articles-2/investigation-in-some-wave-properties-of-a-billiards-cue/?lang=en

Thanks all,
Be well

 

[Jal]

What Rod Cross was studying in the OP.

VII Conclusion

....The experimental data shows that elasticity of the cue tip plays a dominant role in the collision process and suggests that cues with thin shafts might generate lower squirt angles as a result of their greater flexibility rather than their lower mass...

The cue doesn't generate two different forces on the cueball, one from its mass and the other from its stiffness. Its endmass arises from its stiffness. When a cue strikes at dead center, for instance, as it compresses and expands, it exerts forces on the cueball. We don't say the cueball is being propelled by forces generated by the colliding mass of the stick as well as compression/expansion forces. You can look at it either way, but it makes zero sense to add the effects together or say that one contributes more than the other. There's no reason to treat the lateral stiffness/mass any different.

Jim

 

[LAMas]

The cue doesn't generate two different forces on the cueball, one from its mass and the other from its stiffness. Its endmass arises from its stiffness. When a cue strikes at dead center, for instance, as it compresses and expands, it exerts forces on the cueball. We don't say the cueball is being propelled by forces generated by the colliding mass of the stick as well as compression/expansion forces. You can look at it either way, but it makes zero sense to add the effects together or say that one contributes more than the other. There's no reason to treat the lateral stiffness/mass any different.

Jim

Jal,
Hi
If you will, rephrase your position with regards to what is proffered by the two studies and Cornerman's? Are you at odds or in congress with them?

Thanks
Be well

 

[Jal]

If you will, rephrase your position with regards to what is proffered by the two studies and Cornerman's? Are you at odds or in congress with them?

Except for the statement you quoted, I don't think I'm at odds with the studies. (To be honest I just more-or-less skimmed over them.) Fred's said a number of things, but I haven't noticed anything that contradicts my simple contention above.

Your contention is that shaft flexibility has an effect on squirt. I don't think anyone's denied that. But, it "expresses" itself as effective endmass. You can then ask how does altering the stiffness affect that endmass. The answer isn't easy to come by in a detailed quantitative way. But I think you can get a sense of it by comparing the shapes of rods under static loads (which are easier to come by) and observing the shape of the end of a cue during impact from high speed videos. It remains almost straight, indicating that additions or subtractions of endmass as a function of stiffness primarily takes place farther down the shaft than right near the tip. Since the lateral velocity there is small compared to the region near the tip, it seems reasonable to conclude that varying stiffness has only a small effect on squirt.

Sorry if I'm just rambling on about stuff you already agree with. If you don't agree, could you be more specific?

Jim

 

[LAMas]

Except for the statement you quoted, I don't think I'm at odds with the studies. (To be honest I just more-or-less skimmed over them.) Fred's said a number of things, but I haven't noticed anything that contradicts my simple contention above.

Your contention is that shaft flexibility has an effect on squirt. I don't think anyone's denied that. But, it "expresses" itself as effective endmass. You can then ask how does altering the stiffness affect that endmass. The answer isn't easy to come by in a detailed quantitative way. But I think you can get a sense of it by comparing the shapes of rods under static loads (which are easier to come by) and observing the shape of the end of a cue during impact from high speed videos. It remains almost straight, indicating that additions or subtractions of endmass as a function of stiffness primarily takes place farther down the shaft than right near the tip. Since the lateral velocity there is small compared to the region near the tip, it seems reasonable to conclude that varying stiffness has only a small effect on squirt.

Sorry if I'm just rambling on about stuff you already agree with. If you don't agree, could you be more specific?

Jim

I agree.

I like to refer to the mass as "effective mass" that is inclusive of the end mass and stiffness that may contribute to the amount of observed lateral/transverse movement and forces, if any, not to mention the compressing of the tip by the CB.

In the end, this is just semantics if the above is rejected for it seems to be negligible.

The two instrumented scientific studies then were much $$$ ado about nothing. LOL

Be well.

 

[dr_dave]

Your contention is that shaft flexibility has an effect on squirt. I don't think anyone's denied that. But, it "expresses" itself as effective endmass. You can then ask how does altering the stiffness affect that endmass. The answer isn't easy to come by in a detailed quantitative way. But I think you can get a sense of it by comparing the shapes of rods under static loads (which are easier to come by) and observing the shape of the end of a cue during impact from high speed videos. It remains almost straight, indicating that additions or subtractions of endmass as a function of stiffness primarily takes place farther down the shaft than right near the tip. Since the lateral velocity there is small compared to the region near the tip, it seems reasonable to conclude that varying stiffness has only a small effect on squirt.

Jim,

Based on our recent communication, I've add some additional explanations to the endmass resource page in response to some of the points you raised. Since other people seem interested, I've included it below. Please let me know if you think this explanation is still unpalatable.

Regards,
Dave


The forward impulse on the CB is:

F_imp = m_ball * v_fwd

where v_fwd is the forward speed of the CB.

For a given squirt angle "a," the sideways impulse acting on the CB, which acts equal and opposite on the tip, is:

S_imp = F_imp * tan(a) = m_ball * v_side

where v_side is the sideways speed of the CB.

The "effective endmass" feels this impulse, but it also feels an impulse resulting from the force resisting shaft bending because the flexed shaft is pushing back on the endmass (and an equal and opposite force is felt on the remainder of the cue), so the proper momentum equation for the endmass is:

S_imp - F_flex*T = m_end*v_end

where F_flex is the average force associated with the flex of the shaft and T is the tip contact time.

Combining the two previous equations gives:

m_ball * v_side = m_end*v_end + F_flex*T

Therefore, it is clear that the CB's sideways momentum comes from two effects: the momentum transfer from the endmass (m_end*v_end) and the impulse of the flex force (F_flex*T). Effective endmass is affected by lateral or transverse stiffness (as described on the endmass resource page), but force due to flexing is a separate effect. This is clear because you can have one without the other. If most of the endmass were in the tip and ferrule, and the shaft had little or no stiffness (i.e., if it took little or no force to flex the shaft), there would still be the "m_end*v_end" effect but little or no F_flex*T effect. And if the shaft end were stiff laterally but had negligible endmass (even though a greater length of the shaft would contribute to endmass), there would still be a "F_flex*T" effect but little or no "m_end*v_end" effect. When the CB pushes sideways on the tip, it creates endmass momentum, but it also flexes the cue. Both of these things require force, hence the two terms in the equation above.

In the TP B.19 analysis, I am comparing the peak force involved with S_imp to the peak force involved with F_flex. The flex effect is shown to be very small in comparison to the endmass effect.

There is a little "smoke and mirrors" going on here due to the awkward definition of "effective endmass" and because the flex force is actually a dynamic and distributed force acting along the "endmass" and beyond. Also, my static measurement of flex force and deflection doesn't perfectly model the shaft flex involved with the sideways endmass motion, but I think the analysis is reasonably sound to get good ball-park numbers.

 

[ENGLISH!]

^^^^^^^^^^^^^^^^^^^^

 

[LAMas]

More good input from the past. FYI

Originally Posted by Bob Jewett
2008

The stick transfers energy to the cue ball by compressing like a spring along its whole length. The compression wave happens at the speed of sound in the stick, which is about 13000 feet per second. This speed is the fastest that the butt can learn of something colliding with the tip. Some people make the mistake of thinking of the cue stick as being perfectly rigid and incompressible, but it's not. So, the shot proceeds like this: the stick is coming forward and the tip meets the ball. The tip starts to compress, force and acceleration of the cue ball start to build up. The ball also starts to compress, since it too is not incompressible. The ball has started to move, but is not up to the speed of the stick yet, and the stick has started to slow down as its energy is transferred to the cue ball. This continues until the tip (and ferrule and joint and butt) reach maximum compression along the length. At this exact point some amazing things are happening. The stick and ball are moving at the same speed. The force between stick and ball are at their maximum. The compression along the length of the stick (including the tip) is at its maximum. The energy stored in the spring-like compression of the tip (and stick and ball) are at their maximum. For a typical ball and stick, the speeds of the ball and stick are 75% of the original stick speed.

After this point of maximum compression, the ball is pushed forward from the tip by the compression of system. The ball starts to move even faster from this force and the stick continues to slow down. This "unwinding" process continues until the ball finally leaves the tip. At that point, the ball is going at about 130% of the original stick speed, and the stick has slowed down to about 50% of its original speed. (The 130% would be 150%, but the tip is not perfect in springing back to its original shape, and energy is lost.)

Now the hand comes in. Human flesh makes a much "softer" spring than the leather of a tip or the wood that is compressed along the length of the stick. Think of the tip as about the stiffest car spring you can imagine and your hand like a rubber band. The cue ball is gone by the time your hand -- which is still moving forward at full speed -- can wind up even a little. As the hand winds up on the stick and relaxes, which takes about 20 milliseconds, the hand is slowed to about 80% of its initial speed and the stick goes from 50% back up to 80% of its initial speed. Of course this re-acceleration of the stick by your hand is useless in that the cue ball is long gone.

How does a heavier stick affect things? It changes that 130% number. The formula is in Byrne's Advanced book, and somewhere in my columns in Billiards Digest and certainly in Ron Shepard's paper and Dr. Dave's book. A heavier stick through the spring action, puts slightly more energy into the cue ball.

As for how the weight of the stick affects the squirt, I think the answer is that it doesn't, much. Squirt is caused by the spinning cue ball pushing the stick to the side during the contact time of an off-center hit. The amount of squirt is determined by the mass that is being pushed to the side. Since the stick is very floppy side-to-side (as compared to length-wise compression), only the front part of the stick can participate in the squirt during the 1 millisecond or so of contact time. A heavier stick will increase the contact time a little, and that will increase the squirt a little, but I think this effect is pretty small.

Phrased technically, the transverse wave has a very slow propagation velocity along the length of the stick, and so the joint and butt cannot participate in the sideways push that causes squirt.

 

[ENGLISH!]

Good Morning E,

Have you seen the video of Bob Meucci testing the Balabushka Cue made by Helmstetter? https://youtu.be/RIKCiJKsjCQ?t=953


Compared to all of the other cues in his tests it hit the best with the least squirt & he seemed to be a bit shocked & made a comment about how well the butt must have been made. It hit so well in fact that, even though it also hit the best with the Black Dot Shaft, it had the least percentage of improvement with the Black Dot shaft. I am fairly sure that this was the inspiration for the creation of his Power Piston Butt Series of Cues.

I understand the theory that the human hand can not connect to a cue in the same manner as a mechanical robot. I know that it is basically about the mass velocity force. But, the robot connection was the same for all of the cues. I also understand that the bridge of the robot is not exactly the same as a human hand for a closed bridge & that it was not adjusted for different shaft tapers which might result in some small variances as to exactly where the CB might have been hit, but the speed of the shot was certainly fast enough to take swerve virtually out of the equation. So, the squirt direction may not all have been exactly in the precise horizontal direction, but I think given the amounts of squirt represented those differences would be rather small & he was measuring in inches & fractions of inches & not down to the millimeter.

I do not know about everyone else but the points of my hand that connect to the cue do not have much 'meat' between the bones & the shin. I would not compare that area to be much like bubble wrap. I'd compare it to be much more like a very thin piece of tire inner tube where there would be a slight bit of movement if pulled significantly. That said, when making a stroke with the cue, the cue 'slides' a very slight amount to the backward part of the connect & stays tightly there until contact with the ball. To me, it is fairly 'locked' into position & does not have much if any allowance for any further movement during contact other than perhaps being pushed into a tighter connection by the force of the collision with the ball.

I guess what I want to ask YOU, an aerospace engineer... is do YOU... think that the hand's connection to the cue can bring the flex bend of the cue up into the joint & forearm area of the butt so as to have an effect during contact time if not for the 'rebound' of the flex... perhaps for the initial bend.

Now please keep in mind that I'm not talking about a 'grip' of the cue where it is basically just sitting in a 'cradle' with the thumb & curl of the fingers basically just there to keep it from rolling out off where gravity is holding it in the upper fingers & it rocks on different parts of the hand for a full pendulum swing of the cue. I'm talking about a connection where the cue is basically pinched between the two bones of the thumb & index finger & a stroke where the cue is moved in a straight line & the elbow drops so the the hand is in front of the elbow at contact.

I understand how the tight robot connection would add mass but that could certainly be matched or more than matched by an increase in velocity so that the effective force would be equal. I've only put that grip stuff in there for other reasons. Also, I understand how a human body can be bio-mechanically positioned differently to accept an application of force.

Thanks in Advance,
Rick

 

[LAMas]

The Meucci experiment has been debunked about its results but the laws of physics were not. The results were hyperbolic but got some minds (pool player types included) to thinking about the principles involved and how to improve the experiment and that has been discussed since.

It is held by some that the lateral frequency initiated when the tip hits the CB (like a diving board) travels down the shaft and into the grip and can be felt by the grip. By that time, the CB has left the tip.

You say that you use an open bridge with a shallow "V" so I don't think you feel much at the bridge but you do notice that at times, the shaft moves laterally out of your "V". Does that reduce the "effective mass" at the tip...less squirt and more spin?
That might be an interesting experiment to report back on. Do one handed shooters get more spin?

Be well.

 

[ENGLISH!]

The Meucci experiment has been debunked about its results but the laws of physics were not. The results were hyperbolic but got some minds (pool player types included) to thinking about the principles involved and how to improve the experiment and that has been discussed since.

It is held by some that the lateral frequency initiated when the tip hits the CB (like a diving board) travels down the shaft and into the grip and can be felt by the grip. By that time, the CB has left the tip.

You say that you use an open bridge with a shallow "V" so I don't think you feel much at the bridge but you do notice that at times, the shaft moves laterally out of your "V". Does that reduce the "effective mass" at the tip...less squirt and more spin?
That might be an interesting experiment to report back on. Do one handed shooters get more spin?

Be well.

Thanks, E.

I was not actually talking about my bridge hand but what you are talking about here goes back to the swipe motion.

I've played with a guy that plays very well one handed & that is how he always plays even though he has two good hands. I've learned a couple of things watching him play that has helped my one handed shots that I sometimes shoot rather than using a bridge.

He sometimes drove the cue 'straight through' the ball when hitting with english & he at other times had a swiping motion.

Anyway...

My question was more about how much of the cue might actually come into play regarding the 'compression' of the stick.

Do you think that the loading of the 'spring' can be up to the point of the joint & perhaps into the forearm of the stick so that it might have some effect on the contact with the ball? If not in the recoil end, than perhaps in the compression aspect? Either softening of the hit or if restoring itself during contact have another possible effect.

The Power Piston Butts are not solid pieces of wood but are instead smaller diameter pieces of wood encased in a plastic of some kind & then there is the splicing of either wood or in some cases plastic &/or a combination of splicing wood into plastic. A G10 pin is also an option for some on the Meucci Cues. I've noticed that the clear coat on the Power Piston cues cracks in circles at points on the forearm & that tends to tell me that it is flexing & not staying rigid.

We're talking about contact here & during contact time. But as Renfro Chris said the 'feedback' is also a big part as it is part of the subjective learning process for the next stroke & the ones after that, even if the ball is gone by the time that some of us feel the hit & the connection to the ball & subconsciously 'recognize' it even if after the fact that the ball is gone.

Thanks Again & in Advance for any more that you can share...
& You Stay Well,
Rick

 

[Jal]

Jim,

Based on our recent communication, I've add some additional explanations to the endmass resource page in response to some of the points you raised. Since other people seem interested, I've included it below. Please let me know if you think this explanation is still unpalatable.

Regards,
Dave


The forward impulse on the CB is:

F_imp = m_ball * v_fwd

where v_fwd is the forward speed of the CB.

For a given squirt angle "a," the sideways impulse acting on the CB, which acts equal and opposite on the tip, is:

S_imp = F_imp * tan(a) = m_ball * v_side

where v_side is the sideways speed of the CB.

The "effective endmass" feels this impulse, but it also feels an impulse resulting from the force resisting shaft bending because the flexed shaft is pushing back on the endmass (and an equal and opposite force is felt on the remainder of the cue), so the proper momentum equation for the endmass is:

S_imp - F_flex*T = m_end*v_end

where F_flex is the average force associated with the flex of the shaft and T is the tip contact time.

Combining the two previous equations gives:

m_ball * v_side = m_end*v_end + F_flex*T

Therefore, it is clear that the CB's sideways momentum comes from two effects: the momentum transfer from the endmass (m_end*v_end) and the impulse of the flex force (F_flex*T). Effective endmass is affected by lateral or transverse stiffness (as described on the endmass resource page), but force due to flexing is a separate effect. This is clear because you can have one without the other. If most of the endmass were in the tip and ferrule, and the shaft had little or no stiffness (i.e., if it took little or no force to flex the shaft), there would still be the "m_end*v_end" effect but little or no F_flex*T effect. And if the shaft end were stiff laterally but had negligible endmass (even though a greater length of the shaft would contribute to endmass), there would still be a "F_flex*T" effect but little or no "m_end*v_end" effect. When the CB pushes sideways on the tip, it creates endmass momentum, but it also flexes the cue. Both of these things require force, hence the two terms in the equation above.

In the TP B.19 analysis, I am comparing the peak force involved with S_imp to the peak force involved with F_flex. The flex effect is shown to be very small in comparison to the endmass effect.

There is a little "smoke and mirrors" going on here due to the awkward definition of "effective endmass" and because the flex force is actually a dynamic and distributed force acting along the "endmass" and beyond. Also, my static measurement of flex force and deflection doesn't perfectly model the shaft flex involved with the sideways endmass motion, but I think the analysis is reasonably sound to get good ball-park numbers.

Dr. Dave,

Speaking, at best, as your "student," I can't agree with the math for the reasons indicated earlier, as well as in the emails.

But even if the ball experienced the flexing force as a separate and additional entity apart from endmass acceleration, calculations (Euler-Bernoulli treatment) based on an effective cantilever length of, say 6" (as opposed to the entire length of the cue), indicate a force in the area of 2.7 to 3 lbs at a displacement of .038 inches for a maple shaft, depending on whether it's cored out or not. This is, of course, a static figure and may mean next to nothing for the dynamic case (and the 6" is debatable), but it does suggest a much larger number than 1.5% of the total lateral force.

Jim

 

[LAMas]

I don't know.
If you want to try an experiment, buy some Bellville Washers that fit over your pin and don't over tighten the spring. I would be best if you use a loose fitting shaft with a butt joint (wood to wood) and report back if there is a detectable difference.

https://en.wikipedia.org/wiki/Belleville_washer

Be well

 

[ENGLISH!]

I don't know.
If you want to try an experiment, buy some Bellville Washers that fit over your pin and don't over tighten the spring. I would be best if you use a loose fitting shaft with a butt joint (wood to wood) and report back if there is a detectable difference.

https://en.wikipedia.org/wiki/Belleville_washer

Be well

Thanks, E.

I have a cue that I can do that. I don't know when I will next be at Home Depot, But I will keep this in mind.

Thanks Again,
Rick

PS Edit: I'm not sure that an 'experiment' with that washer would really tell me anything really as it is only taking out or disrupting the lateral transmission other than through the connecting pin & an area immediately surrounding it, really. I sort of see the intention but it would just be a different type of 'distribution' or transmission. Perhaps just a simple rubber washer might be more telling.

 

[dr_dave]

You say that you use an open bridge with a shallow "V" so I don't think you feel much at the bridge but you do notice that at times, the shaft moves laterally out of your "V". Does that reduce the "effective mass" at the tip...less squirt and more spin?
That might be an interesting experiment to report back on. Do one handed shooters get more spin?

A human bridge (regardless of the type) has no effect on effective endmass or squirt (for typical humans and typical pool equipment). Here's a pertinent quote from the squirt bridge length effects resource page:

Does the bridge length or tightness have any affect on squirt (cue-ball deflection)?

No, unless the bridge length is really short and the bridge fingers are very bony and have an extremely tight (i.e., non human) grip around the cue.

Even if the bridge were perfectly rigid, it would still have absolutely no effect for bridge lengths beyond about 6-8 inches. The following video (at the 2:32 point point in part 2) shows and explains why visually: NV B.96 - Grip and bridge technique and advice.

And Diagram 4 in the following article gives some additional experimental proof related to endmass:

"Squirt - Part VII: cue test machine results" (BD, February, 2008)

Enjoy,
Dave

 

[dr_dave]

Dr. Dave,

Speaking, at best, as your "student," I can't agree with the math for the reasons indicated earlier, as well as in the emails.

But even if the ball experienced the flexing force as a separate and additional entity apart from endmass acceleration, calculations (Euler-Bernoulli treatment) based on an effective cantilever length of, say 6" (as opposed to the entire length of the cue), indicate a force in the area of 2.7 to 3 lbs at a displacement of .038 inches for a maple shaft, depending on whether it's cored out or not. This is, of course, a static figure and may mean next to nothing for the dynamic case (and the 6" is debatable), but it does suggest a much larger number than 1.5% of the total lateral force.

Jim,

That's a good point concerning the effective length of the flex during tip contact. I agree with you that it would be shorter than in a static deflection test. Although, with the Predator Z-2, the tip end of the shaft is very flexible with its small diameter (11.75mm) and hollow core (about 5 inches long), so I suspect a lot of the flex in the static case also occurs closer to the tip than the joint. When I get a chance, I'll do the static flex test again and trace the shape of the flexed shaft.

Regards,
Dave

 

[Corwyn_8]

a force in the area of 2.7 to 3 lbs at a displacement of .038 inches for a maple shaft, depending on whether it's cored out or not.

This corresponds to DrDave's "transverse stiffness", yes?

So you are calculating a value of 71 lbf/in
And DrDave is measuring a value of 2 lbf/in

That would certainly explain your different expectations.

Can you measure a few cues?
Perhaps DrDave will replicate your calculations?

And we will have our resolution.

Thank you kindly.

 

[Mr. Bond]

F_imp = m_ball * v_fwd

where v_fwd is the forward speed of the CB.

For a given squirt angle "a," the sideways impulse acting on the CB, which acts equal and opposite on the tip, is:

S_imp = F_imp * tan(a) = m_ball * v_side

where v_side is the sideways speed of the CB.

...so the proper momentum equation for the endmass is:

S_imp - F_flex*T = m_end*v_end

where F_flex is the average force associated with the flex of the shaft and T is the tip contact time.

Combining the two previous equations gives:

m_ball * v_side = m_end*v_end + F_flex*T

bake at 375F for 35-40 mins or until golden brown

cool for 10 minutes

slice and serve immediately

 

[dr_dave]

Yummy! :thumbup:

F_imp = m_ball * v_fwd

where v_fwd is the forward speed of the CB.

For a given squirt angle "a," the sideways impulse acting on the CB, which acts equal and opposite on the tip, is:

S_imp = F_imp * tan(a) = m_ball * v_side

where v_side is the sideways speed of the CB.

...so the proper momentum equation for the endmass is:

S_imp - F_flex*T = m_end*v_end

where F_flex is the average force associated with the flex of the shaft and T is the tip contact time.

Combining the two previous equations gives:

m_ball * v_side = m_end*v_end + F_flex*T

bake at 375F for 35-40 mins or until golden brown

cool for 10 minutes

slice and serve immediately

 

[Jal]

This corresponds to DrDave's "transverse stiffness", yes?

Yep.

So you are calculating a value of 71 lbf/in
And DrDave is measuring a value of 2 lbf/in

Yes, about that (77 lb/in), although my number is based on a 1/2" shaft, a bit thicker than Dr. Dave's shaft. Dr. Dave's measurement was with the cue supported back in the butt region, so most of the cue was flexing, whereas the calculation I did was based on a cantilever length of only six inches (because of the slow propagation of the shear wave). The "spring constant" is inversely proportional to the cube of the cantilever's length, so that's where the major difference comes in. Also, since Dr. Dave's measurement involved the tapered portion of the cue, differences arise from that as well, but would tend to bring the numbers closer together since that portion is much stiffer than the untapered section. The chance of these static measurements/calculations reflecting reality for the dynamic case is not all that great, imo.

That would certainly explain your different expectations.

Can you measure a few cues?

I only have one. Dr. Dave may redo his so I might just wait on that. I would like to compare his previous measurement with a calculation using the entire length of the cue as a check on the math, but, as I derived (modified) the math for the tapered section, I'd feel better if I went through the theory and derivation again . Being very rusty, this could take some time (it all looks like Greek to me at this point).

And we will have our resolution.

I'm afraid our differences are more basic.

Jim