A ball struck by a cue in billiards with English goes straight at first....

[AstroChris]

Wow. This got to all kinds of new levels of science to describe this game. Truthfully, the only academics you really need to know is Geometry and Trigonometry. Dead honest truth right there. Knowing the physics of why the ball is reacting the way it is after being struck is, in all aspects, a ludicrous question in its plain form because there are simply going to be way, way too many variables. You have to take your stroke, your bridge, the air density and temperature, the cloth, the table (level), ball size, ball weight, definitions of terms and so much more out of the equation by making them the control. It's tough to break this down per physical law, it just is.

 

[poolplayer]

Knowing the physics of why the ball is reacting the way it is after being struck is, in all aspects, a ludicrous question in its plain form because there are simply going to be way, way too many variables.

Yeah... but it is just a ball hit by a stick... I am not saying that I want to predict the ball direction in millimeter precision. I am just wondering what force makes the cue ball go straight with right English. We know the ball's size and weight. That ball is relatively heavy, so I don't think air would affect significantly the ball direction. We can shoot the ball straight even in summer or in winter so the temperature or air density is not relevant in normal condition. Stroke has to be straight, but bridge is not so relevant. Even without bridge, I can strike the cue ball straight with right English.

I thought that physics savvy could come up with several key relevant forces and simple equations, but maybe this question was tough one or just hard to explain with plain words. But, I have learned several ideas here. Probably friction would be the key player as Cutter Ketch and rcgldr suggested. I will learn some physics little by little until I can be sure of that. Thanks everyone, and have happy holidays!

 

[A.T.]

even if there is zero friction, if the cue strikes the ball off center, the ball will spin

If there is zero friction, there is only a normal contact force, that passes through the ball's center of mass. How would that make it spin?

 

[A.T.]

Imagine the ball rotating anticlockwise with its axis vertical. Now tilt it forward (as when you give it any forward motion with the cue). The bit in contact with the ground will be pushing against the ground to the left, which will result in a force steering the ball to the right. That's where your rightwards force comes from.

Here is more on this:

http://www.physics.usyd.edu.au/~cross/PUBLICATIONS/55. RollingBall.pdf

The whole thing requires you to think in 3D and to consider two axes of rotation.

Yes, rolling while spinning around the normal axis can be quite counter intuitive:

 

[rcgldr]

even if there is zero friction, if the cue strikes the ball off center, the ball will spin. Consider what would happen if a rod in space was struck near the end, the rod would end up rotating in addition to translating due to the collision.

If there is zero friction, there is only a normal contact force, that passes through the ball's center of mass. How would that make it spin?

The ball is struck off center so the force vector does not pass through the ball's center of mass, but to the right side of the center of mass (in the right spin example case used in this thread), resulting in a torque about the vertical axis that causes the ball to spin.

 

[sophiecentaur]

even if there is zero friction, if the cue strikes the ball off center, the ball will spin

No! Not if there is no friction. Your Physics / intuition is letting you down here. No friction means that there can be no tangential force on the ball and the only force can be radial.

the off center forward force imparted by the cue onto the ball would generate a component of friction force to the right onto the cue

Force on cue is rightwards. Force on ball is leftwards. You have to be strict with your force direction signs. Have you done the more simple sums describing ball - ball interactions? The impact will cause the balls to separate - same thing happens with cue and ball. The ball cannot 'follow' the cue.
You are relying on your intuition and it is letting you down. It doesn't matter but just don't pretend that this discussion involves Physics.

 

[rcgldr]

Your Physics / intuition is letting you down here. No friction means that there can be no tangential force on the ball and the only force can be radial.

Force on cue is rightwards. Force on ball is leftwards. You have to be strict with your force direction signs. Have you done the more simple sums describing ball - ball interactions? The impact will cause the balls to separate - same thing happens with cue and ball. The ball cannot 'follow' the cue.
You are relying on your intuition and it is letting you down. It doesn't matter but just don't pretend that this discussion involves Physics.

I struck out my prior comment. Friction is needed.

 

[sophiecentaur]

I think we are getting there. That link about golf, pool and bowling says it all.

 

[poolplayer]

Hi, I realized that the difficulty of this problem is that we actually don't know what is happening during collision in detail. So, I decided to collect more data. Please check a video showing that cue ball direction can be almost normal when using a heavy hard stick.

It has been known that the cue ball deflection increases (cue ball does not go straight) when we increase the cue tip density (mass). I put a normal tip to a relatively heavy aluminum stick and shot the ball. The resultant direction was indeed near normal from the contact surface. Dr. Dave says that the effective cue tip mass increases the cue ball deflection (squirt) through bouncing force to the ball after cue bend without mentioning exactly what force makes the cue ball go straight (he did mention friction force on the ball but he didn't note its direction saying it is difficult to calculate).
http://billiards.colostate.edu/technical_proofs/new/TP_A-31.pdf

My current hypothesis is that friction force through rightwards cue deflection makes the cue ball go straight. By using a heavy cue, the speed of the rightwards cue deflection decreases, so the rightwards friction force (like when a ball is contacting a conveyor belt running to the right) decreases and the ball goes to normal from the contact surface.

Anyway, this video clearly shows that the spin/rotation or friction between the ball and table is not the key factor that makes the ball go straight. I believe that the spin affects the cue direction only after slipping phase, not during collision.

 

[poolplayer]

I think the friction is rightwards because the cue slides rightwards on the ball. Friction force seems less than the product of friction coefficient and normal force to the cue if the ball and cue are moving together. But, I still can't figure out this diagram can explain that a ball goes straight...

 

[poolplayer]

I still can't figure out this diagram can explain that a ball goes straight...

Maybe in this diagram the ball goes to normal direction if the cue does not deflect (if cue movement can be fixed straight without bend)...

 

[Cutter Ketch]

Any force that rotates a ball isolated in space can't affect the direction it heads off in. That direction of the linear momentum change can only be radial from the contact point.

That statement is incorrect. You must satisfy BOTH Γ = I α and F = m a simultaneously. The free body will accelerate in the direction of the applied force even a tangential one.

Picture a circular puck on an air table. The puck has some thread wrapped around it. Pulling on the thread the force is purely tangential. You can't pull the thread without the puck moving toward you. The direction the puck moves is directly toward you. The line of action relative to the center of mass doesn't matter. The acceleration is in the direction of the force. F=ma

When the cue strikes the ball you apply a force in the direction the cue is pushing. There is no surprise that that is the direction the ball moves. The only question is how the tangential component is imparted, and the answer is friction. The rest of the discussion has been about interesting perturbations like when the cue bends or friction fails.

 

[sophiecentaur]

That statement is incorrect. You must satisfy BOTH Γ = I α and F = m a simultaneously. The free body will accelerate in the direction of the applied force even a tangential one.

Picture a circular puck on an air table. The puck has some thread wrapped around it. Pulling on the thread the force is purely tangential. You can't pull the thread without the puck moving toward you. The direction the puck moves is directly toward you. The line of action relative to the center of mass doesn't matter. The acceleration is in the direction of the force. F=ma

When the cue strikes the ball you apply a force in the direction the cue is pushing. There is no surprise that that is the direction the ball moves. The only question is how the tangential component is imparted, and the answer is friction. The rest of the discussion has been about interesting perturbations like when the cue bends or friction fails.

You're right. That statement of mine doesn't make a lot of sense on its own. I haven't read the context of that sentence so I'll have to let it ride, except to say that there is no tension involved in pool and the only forces that can act will be radial and tangential.
Having read that link about pool and bowling, which gives the answer to all these questions in a succinct way, I can't understand why anyone continues to argue about the way the whole thing works. People should read that site carefully and not skip the bits that they don't understand. No one plays pool on a frictionless table or with a frictionless cue tip. The thing that moves the ball to the right is the forward tilt of the ball as it rolls, combined with the spin due to the off centre cue contact.

 

[poolplayer]

The only question is how the tangential component is imparted, and the answer is friction. The rest of the discussion has been about interesting perturbations like when the cue bends or friction fails.

Yes, that should be friction. I am now totally certain after seeing your discussions and doing some experiments.

Last thing I want to add is that maybe the cue bend is the main factor that determines the ball direction. This time I set my bridge (fixing point of the cue) very close to the ball and stroke it. With the near bridge, the cue could not flex much as before. And as result, the ball went near normal direction.


My guess is that the near bridge decreased the speed of initial cue deflection/flex to the right and reduced the friction on the ball. Due to this reduction of the rightwards friction, the ball went normal from the contact surface. Although I am not quite sure how much the rebound of the cue to the left affected the cue direction, it seems that the ball already started to run oblique during the initial cue flex to the right.

This finding was a little bit surprising to me because it has been reported that cue flex does not affect physics of cue ball much. But, considering it happens during contact with the cue ball, it is reasonable that the ball follows the cue flex movement as long as the force is in the range of static friction.

 

[poolplayer]

Having read that link about pool and bowling, which gives the answer to all these questions in a succinct way, I can't understand why anyone continues to argue about the way the whole thing works. People should read that site carefully and not skip the bits that they don't understand. No one plays pool on a frictionless table or with a frictionless cue tip. The thing that moves the ball to the right is the forward tilt of the ball as it rolls, combined with the spin due to the off centre cue contact.

We all know that the spin changes the ball direction, but this happens after the collision and not during collision.

You know that a ball can go almost the same direction regardless of different friction coefficient of the table.
1) usual table:

2) smoother table:

(there is still friction but curve should be reduced than a usual table)

And you know that a ball can go oblique even if the ball is spinning.


I don't know why you still believe that spin is important for the cue ball direction shown in these video. First of all, the ball trajectory in these videos is quite much linear...
The link was interesting showing the change of the spin axis during rolling of a ball, though.

 

[sophiecentaur]

A point about cue deflection. The more flexible the cue is, the longer it will stay in contact with the ball. This means the spin around a near vertical axis will be greater because the horizontal force has been acting for longer.

 

[A.T.]

Note how the ball rotates, during and right after the impact. It's not just spinning around the vertical axis. Hence I don't think you can treat this as a simple 2D problem. For example: the contact force from the cue tip could have a downwards component, which presses the ball into the table and "tilts" to the right.

 

[poolplayer]

The more flexible the cue is, the longer it will stay in contact with the ball. This means the spin around a near vertical axis will be greater because the horizontal force has been acting for longer.

This is a good point. My understanding is that the tangential force to the right induces both rightwards drive and counterclockwise spin, so it would be hard to dissociate these variables experimentally. When I see my videos, I feel that the contact time is actually shorter with a flexible cue, but I am not sure about that...

Note how the ball rotates, during and right after the impact. It's not just spinning around the vertical axis. Hence I don't think you can treat this as a simple 2D problem. For example: the contact force from the cue tip could have a downwards component, which presses the ball into the table and "tilts" to the right.

I agree that 3D information would be necessary to explain details of the long distance ball trajectory, but I think this is not necessary to answer my question because my 2D diagram seems sufficient to explain the coarse ball direction immediately after the collision.

I noted this in youtube, but I intentionally hit slightly below the mid height on the right hand side of the cue ball in the video. I am sure that the ball rotates only around the vertical axis if I hit the mid height. Of course the vertical axis would be maintained only a short period of time after the collision when the ball is still slipping on the table. With a sufficient friction between the ball and table, the axis should soon change as shown in your link. And the ball should "tilt" to the right while the ball rolls, although this could not be observed in the short video.

 

[sophiecentaur]

because my 2D diagram seems sufficient to explain the coarse ball direction immediate after the collision.

Not at all. The whole roll and spin is important if you want to predict what happens. Without the contact with the table there is nothing to give the ball a rightwards force to counter the net leftwards force from the cue. You cannot produce a rightwards force by just waving metaphorical arms about. You need to identify an actual force on the ball which will achieve it. You have acknowledged that the cue moves to the right so the force on the ball (a reaction force from the cue) must be to the left.

I am sure that the ball rotates only around the vertical axis if I hit the mid height.

But doesn't the ball roll forward? You cannot think it just skids along.

 

[poolplayer]

Without the contact with the table there is nothing to give the ball a rightwards force to counter the net leftwards force from the cue. You cannot produce a rightwards force by just waving metaphorical arms about. You need to identify an actual force on the ball which will achieve it. You have acknowledged that the cue moves to the right so the force on the ball (a reaction force from the cue) must be to the left.

Are you sure that the friction between the ball and table is necessary for the tangential force? You keep me confused... Wait, do you think that the tangential force does not exist?

But doesn't the ball roll forward?

It will roll forward after it stops slipping on the table. The ball rotates only around the vertical axis while it is slipping after shot in the mid height with right English. I think you can see it from some of my videos...

 

[sophiecentaur]

re you sure that the friction between the ball and table is necessary for the tangential force? Y

No, the tangential force between cue and ball is there whenever contact is not exactly normal. That tangential force will spin the ball about a near vertical axis.

It will roll forward after it finish slipping.

It may take a short time to start rolling forwards at full speed but even if it does slip against the table (and there is no reason why it should slip at all at some contact heights) it will be rolling almost instantly.
Your argument about using 2D analysis implies that you could get a straight shot with a penny, hit off axis with a wooden ruler. You could even try that on a laminated table top and see if you can actually achieve. I see no reason that it could happen.

 

[poolplayer]

No, the tangential force between cue and ball is there whenever contact is not exactly normal.

Then, the friction between the table and ball is not necessary for the tangential force...

It may take a short time to start rolling forwards at full speed but even if it does slip against the table (and there is no reason why it should slip at all at some contact heights) it will be rolling almost instantly.

A ball can slip in a long distance probably up to 10 inches. This is what all pool players know.

Your argument about using 2D analysis implies that you could get a straight shot with a penny, hit off axis with a wooden ruler. You could even try that on a laminated table top and see if you can actually achieve. I see no reason that it could happen

I didn't understand this example...(penny put on the tip of a wooden ruler??) but particularly what force are you against in the diagram?

 

[poolplayer]

No, the tangential force between cue and ball is there whenever contact is not exactly normal. That tangential force will spin the ball about a near vertical axis.

Oh, maybe you think that tangential force only makes a ball spin and does not push ball to somewhere.

 

[poolplayer]

I didn't understand this example...(penny put on the tip of a wooden ruler??)

Now I know what you were asking. I think it is possible that a penny goes straight if 1) there is significant friction between the penny and the wooden ruler, 2) the wooden ruler is extremely flexible so that it can be flexed in high speed by the light weight of the penny, and 3) the penny and wooden ruler are in contact during the wooden ruler's flex.

 

[poolplayer]

I think I have got an answer in this thread, so here I summarize the conclusion.



1) It is rightwards friction between the cue and ball that makes the ball go straight (rather than oblique to the left) with right English. To my surprise, probably there is no reason that the ball must follow the cue direction with English. One of the reason that the ball goes almost straight would be that cue makers refined several cue features so that the ball follows the cue direction.

2) Speed of cue flex during contact between the cue and ball would be the most important feature that makes a ball go straight. When I use a heavy/hard cue or when a bridge (fixing point of a cue) is very near from the ball, the cue cannot flex much. If the cue cannot flex in high speed during the contact, the ball cannot receive much rightwards friction force from the cue, so the ball deflects to the left, closer to the direction normal from the contact surface. Moreover, the frequency of the cue vibration is high when shot with a heavy/hard cue or a bridge near from the ball. This fast vibration could make the cue ball goes more oblique to the left because the rebound of the cue to the left after rightwards flex can push back the ball to the left (this can be double-hit if the cue once leaves the ball after the rightwards flex).

3) Counterclockwise spin induced by right English can make the ball curve to the right after collision. Although this happens after a while after collision when the ball stops slipping on the table, the spin makes the ball walk rightwards, so helping the ball go straight if the ball deflection to the left is significant and there is a long distance to the aiming point.

Tips to pool players:
There is not much to learn here to improve your game. But, the physics of cue-to-ball collision suggests that it is really tricky to make a cue ball run to the cue direction with right/left English. So, it would be wise not to use English as much as possible (everyone knows this though...). And to minimize the cue ball deflection, it would be better to keep your bridge sufficiently distant from the ball. This comes to a problem when the cue ball is very close to the cushion. English would be the last option if you have to shoot a ball frozen to the cushion.

 

[A.T.]

I am sure that the ball rotates only around the vertical axis if I hit the mid height.

Do you have a clip of that? How does the ball move then?

 

[poolplayer]

Here you go. You can find more clips in Dr. Dave's website.
http://www.billiards.colostate.edu/high_speed_videos/new/HSVA-8.htm
I said this many times, but a ball can slip on the table because the friction between the ball and table is not high enough.

 

[sophiecentaur]

It's worth noting that the Force F_ft may be significantly large, as the diagram shows but it is not acting on the cm of the ball and its effect on final direction is tiny, compared with the component that acts directly through the cm.
Lay a ruler on the table and hit the end with an impulse, normal to the line of the ruler. It will rotate a lot more than it will move in the direction of the hit. That diagram suggests otherwise because it is tempting to consider that the resultant force is relevant. Unless the cue tip has adhesive on it, the result motion of the ball can never be in the direction of the impulse from the cue.

I said this many times, but a ball can slip on the table because the friction between the ball and table is not high enough

This is a non linear effect. For a small applied force, there will be no slip. For a large force, there may be a lot of slip and the 2D situation applies more nearly. But any forward roll will make the ball walk to the right as the spin axis is tilting forward.

 

[poolplayer]

It's worth noting that the Force F_ft may be significantly large, as the diagram shows but it is not acting on the cm of the ball and its effect on final direction is tiny, compared with the component that acts directly through the cm.
Lay a ruler on the table and hit the end with an impulse, normal to the line of the ruler. It will rotate a lot more than it will move in the direction of the hit. That diagram suggests otherwise because it is tempting to consider that the resultant force is relevant.

How did you calculate cm change by the Force F_ft to conclude it is tiny?

 

[A.T.]

It's worth noting that the Force F_ft may be significantly large, as the diagram shows but it is not acting on the cm of the ball...

Completely irrelevant. If that force is significant, then it causes significant linear acceleration of the ball in that direction.

it is tempting to consider that the resultant force is relevant.

Because it is, according to Newtons 2nd Law.