Coriolis effect example: ball tossing on rotating carousel

FranzDiCoccio
Hi all,
I was reviewing the Coriolis effect and I came across the attached explanatory image (from the Italian version of a book on physics by Cutnell, Johnson, Young and Stadler).

The idea is the following.
• two guys are facing each other on a rotating carousel;
• one of the guys on the throws a ball towards the other guy;
• of course the ball goes through a straight line;
• of course the guys on the carousel see the ball deviating according to Coriolis effect;
What really does not convince me is the trajectory of the ball in the rest reference frame (red straight dashed line). Would it really go through the center of the carousel?

Here the assumption is that the ball is thrown towards the center (and hence the other guy) in the rotating reference frame. But since the ball is in the hands of the first guy, who is rotating, the ball has a tangential velocity too. If the guy simply drops the ball, this tends to move along the tangent of the circle it was orbiting on.
If I introduce two cartesian axes I can say that the initial velocity of the ball has both a component along x (the guy throws it) and along y (tangential velocity). Therefore it cannot possibly go through the center of the carousel, can it?
I think the authors are simply disregarding the centrifugal force (but they won't mention this).
I guess that the dashed straight trajectory would work for a ball thrown by someone standing in the rest frame behind the first guy.

Personally, I would prefer an example where the ball is thrown from the center of the carousel, so that there is no tangential velocity.

I'm pretty sure this figure is wrong, but I'd like to make sure I'm not forgetting something.
Thanks a lot for your insight.

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Hi all,
I was reviewing the Coriolis effect and I came across the attached explanatory image (from the Italian version of a book on physics by Cutnell, Johnson, Young and Stadler).

The idea is the following.
• two guys are facing each other on a rotating carousel;
• one of the guys on the throws a ball towards the other guy;
• of course the ball goes through a straight line;
• of course the guys on the carousel see the ball deviating according to Coriolis effect;
What really does not convince me is the trajectory of the ball in the rest reference frame (red straight dashed line). Would it really go through the center of the carousel?

Here the assumption is that the ball is thrown towards the center (and hence the other guy) in the rotating reference frame. But since the ball is in the hands of the first guy, who is rotating, the ball has a tangential velocity too. If the guy simply drops the ball, this tends to move along the tangent of the circle it was orbiting on.
If I introduce two cartesian axes I can say that the initial velocity of the ball has both a component along x (the guy throws it) and along y (tangential velocity). Therefore it cannot possibly go through the center of the carousel, can it?
.

Why is that particular trajectory forbidden? Why can't the y component be 0?

If there were a target in the middle of the carousel, do you think that given practice you could hit it from the edge of the moving carousel?

FranzDiCoccio

I'm not saying that hitting the middle of the carousel is impossible. You have simply to compensate for the tangential velocity.
With a little practice you can even get to your friend in front of you.
However, the book does not say that the guy aims a little off the center, to compensate for the rotation so that the ball goes through the center of the carousel.
And anyway, what would be the point of such an example? The guy obviously knows the Coriolis force, but for some unclear reason he goes for the center of the carousel instead of going all the way and compensate so that the ball gets to his friend?

In the example the guy does the obvious, naive thing. Somehow he's doing this for the first time, and he throws the ball towards his friend, who is directly across him, on the other side of the carousel. In my opinion this means that the y component of the velocity is non zero, and the ball does not go through the center of the carousel, as depicted.
Do you agree?
Or likewise, do you agree that the book should say that in order to get the dashed straight trajectory in the figure the guy should aim a little to the right of the center, so that the initial velocity of the ball has a negative y component that cancels the tangential velocity exactly?

I'm not sure what is happening here. It might be a translation error. I'd be curious to read the original text, assuming that this part was not introduced by the Italian editors.

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I'm not saying that hitting the middle of the carousel is impossible. You have simply to compensate for the tangential velocity.
With a little practice you can even get to your friend in front of you.
However, the book does not say that the guy aims a little off the center, to compensate for the rotation so that the ball goes through the center of the carousel.

It doesn't say he doesn't "aim" off centre either. It just shows a perfectly possible straight path for the ball.

I'm pretty sure this figure is wrong,
Impossible to say without the actual caption and description what it is supposed to show. Definitely not the same scenario from different frames, because the ball hits different parts of the carousel.

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Impossible to say without the actual caption and description what it is supposed to show. Definitely not the same scenario from different frames, because the ball hits different parts of the carousel.

Yes, on the second sequence they haven't curved the path enough after it goes through the centre. It should hit the black mark to concur with the first sequence.

FranzDiCoccio
FranzDiCoccio
Yes, on the second sequence they haven't curved the path enough after it goes through the centre. It should hit the black mark to concur with the first sequence.

Yes, well spotted! I thought the bottom sequence looked strange too, but I wanted to understand the top one first. Anyway, I agree with you.

It doesn't say he doesn't "aim" off centre either. It just shows a perfectly possible straight path for the ball.

I'm not sure I understand your point. The path is definitely possible, but even if you are right, and the book does not say anything about the initial velocity, it should!

Anyway I have read again what the book says. The caption gives no clue as to the direction of the velocity of the ball. A rough translation of the main text is
Imagine sitting on a merry go round, facing your friend, and set the merry go round into a fast rotation. Now throw the ball towards your friend, so that he catches it...
I can attach a snapshot of the text, if you like.

In the first snapshot "towards your friend" means along the dotted red path (let us call it the x direction).
Perhaps you can argue that the clause "so that he catches it" implies that you somewhat try to compensate for the rotation, and so you could be right, but the book is not clear: is it "towards your friend" or "slightly off"?

Also, assuming you are right, the example seems a bit odd. You are a student who very likely does know anything about the Coriolis effect, and never tried this experiment in real life. I'm assuming that the average student imagines throwing the ball straight ahead.

Maybe the book could say: "since you're kind of smart, you anticipate that your friend would not be at the same position when the ball reaches the opposite side of the merry go round, and you toss the ball slightly to the right of the center". But It does not say that.
As I mention, I would not choose that particular example, which introduces an unnecessary complication. In order to make things simpler the first boy could be at the center of the platform, so that the ball has no tangential velocity. This is actually what is done in this nice YouTube video. In the second example the guy is on the edge of the merry go round, but the comment to the video clearly states that the ball now has a tangential velocity.

I'm not trying to prove I'm smarter than the authors. Mistakes happen.
I'm just pointing out that either the figure is plainly wrong, or its explanation is unclear and misleading. Either way that's a mistake.
Imagine what a student can make of that explanation. Add to that the mismatch of the two sequences you noticed. I anticipate very unphysical scenarios.

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Perhaps the book expects you to be smart enough to work this out for yourself!

I agree that if it was my book I would have pointed this out. Not least because a number of people get confused on this very point.

Now that you have worked it out, you should move on.

FranzDiCoccio
FranzDiCoccio
I agree that if it was my book I would have pointed this out. Not least because a number of people get confused on this very point.

Exactly. I have to explain the content of this chapter of this textbook to someone who uses it at school. I did study this effect a as a student, but this was a while ago, and I remembered that there are a number of subtleties one should be wary about.

Perhaps the book expects you to be smart enough to work this out for yourself!

Yes, perhaps. I doubt that, though. These are 16 years old. Also, a book can have you guess, but at some point there should be a clear explanation, for you to make sure that your guess is right.

I think that in this case the authors/editors/proofreaders have been a bit shallow.

My personal guess is that the authors made a mistake, because mistakes happen. Then the editors probably hired poor proofreaders, if any.
Or else the proofreaders were good, but they were paid too little to do a really thorough work, and they got sloppy.
You would not believe how many mistakes you can find in these books. Most of them are in the solutions of the problems, but some (as this one) affect the actual explanation of the phenomenon.
I think that pointing them out is useful.

Franz

FranzDiCoccio
Impossible to say without the actual caption and description what it is supposed to show. Definitely not the same scenario from different frames, because the ball hits different parts of the carousel.

Thanks a lot, I did not notice that at first. I focused on the text and the top row, which did not convince me, and I did not look at the bottom one.
Sorry, I got your answer wrong the first time I read it. My English sometimes fails me. Now I understand it.

Here the book is definitely wrong, because from the captions it is clear that this is supposed to be the same scene as seen from two different frames of reference.

[my translation]
Top view of two boys playing a game of catch on a small merry go round spinning at a constant angular velocity.
A) In the frame of reference of Earth you see the ball following a straight line at a constant speed.
B) In the non inertial frame of reference of the merry-go-round the ball follows a curved trajectory.

I see from the second row that the boy actually aims to his right. So confusing.
The text is unclear and the figures are not consistent.

I think that the book should be corrected choosing between the two following options
• stress in the text or the captions that the guy is aiming to his right, and change the second row so that it is consistent with the first;
• have the boy do the naive thing, aiming at his friend, and change both figures accordingly.

FranzDiCoccio
Hi

me again, apologies.
I'd like to understand two further points concerning the Coriolis effect

1) what is the teaching value of the example where the trajectory of the ball goes through the center of the merry go round, if any? It is a very particular one among infinitely many...
2) The book by Cutnell I have writes the Coriolis acceleration as $a_{\rm C}=2 \vec{v}\times\vec{\Omega}$, which is how I remembered it. However I see that many sources use $a_{\rm C}=-2 \vec{\Omega}\times\vec{v}$, which of course is equivalent. However I really dislike that minus. Is there a deep reason for emphasizing a negative sign, or is it just a matter of tradition?