1. The problem statement, all variables and given/known data
I was trying to learn about this force, and came across a youtube video: . At 4.00 in the video, he says, "if the parcel of air continues to get deflected..." Why would it continue to get deflected?
At 4.40 in the video, a girl rolls a ball while moving on a merry go round, and the ball ends up forming a circle and coming back to her. Why does this happen? 2. Relevant equations
I think I'm supposed to use conservation of angular momentum here. So MVR+I*omega is constant. 3. The attempt at a solution
I don't understand how the parcel could continue to get deflected and end up changing direction, and forming a circle. I thought that coriolis effect was observed by people on earth because air packets in motion on the equator are faster than those closer to the poles, so moving towards the poles, they would have a higher speed than the earth there, and thus appear to get deflected towards the east. But once reaching there, do they get deflected back to the equator, like shown in the video? If so, how do they gain the velocity towards the equator? Is it because of conservation of angular momentum, because they have a higher velocity with a lower radius now and need to increase the radius again, and so they move towards the equator?

As for the girl on the merry go round, she rolls the ball, and with respect to the merry go round, it rotates in a circle. How does that happen? Is it due to conservation of angular momentum, again? And how to prove that its a perfect circle?

Well although it is quite confusing, I will try to offer my two cents on the matter, and I remember the lab for this section being of the most fun and conceptually difficult at the time.

For your first question, the reason why the parcel continues to be reflected is because the Coriolis force applies... well a force on the Parcel, therefore the parcel accelerates. So as the parcel goes away from the axis of rotation, it slows down, then accelerates towards the the axis of rotation and speeds up. Does this sound familiar? It should because this is very similar to a centrifugal force, which if you are an observer on earth it seems as though there is this "mystery force" is making the parcel go in circles. (Think of centripetal force and centrifugal force)

In reality the parcel is just being affected by how fast earth rotates right? It will continue to be deflected, but from space it will not appear this way, the parcel will continue to move forward. Try drawing out the scenario as an observer on the earth and as an observer in space (this is simpler than you think, and is easier then your next question).

For the next question, to understand you should try to draw two diagrams. One where the ball is rotation back to the girl (do a frame-by-frame type of diagram) and one where an eagle flying over them will view it. (Eagle-eye type of view) and you will see that the ball does in fact go straight the whole time. (drawing a free body diagram for the ball will immensely help to see the Coriolis force). From there you will see why the ball acts the way it does. Try it out!

The merry go round is the easy case. The Coriolis effect on the surface of the earth is complicated by the fact that we are working in three dimensions and the fact that the atmosphere is bound so it stays close to the surface of the earth. Recall that the Coriolis force scales with the rotation rate (fixed at approximately one rotation per day), the velocity of the object and the sine of the angle between the velocity and the axis of rotation. The direction of the deflection be at right angles to both the axis and the object's original velocity. Instead of considering the sine of the angle, one can instead just consider the component of the object's velocity perpendicular to the axis of rotation and ignore the component (if any) that is parallel to the axis.

For a parcel of air moving northward (for instance), part of the original motion is parallel to the Earth's axis (toward the pole star -- up and north) and part is inward (toward the Earth's axis -- down and north). The component toward the pole star is irrelevant. The component toward the Earth's axis is the important bit. It results in an eastward deflection.

For a parcel of air moving eastward (for instance), the entire eastward motion is at right angles to the axis of rotation. This results in a deflection away from the Earth's axis -- up and south. The upward deflection is essentially irrelevant. The forces of gravity and buoyancy will control where the wind blows vertically. The Coriolis force tends not to matter. What remains is the southward component.

One can repeat the analysis for southward moving air, westward moving air and for any angle in between. There will always be a deflection clockwise (in the Northern hemisphere). The relevant component will always scale with the sine of the latitude.

The circular path traced out by a parcel of air under a Coriolis acceleration has remarkably little to do with the girl on the merry go round throwing a ball in a circular path...

If (and only if) she rolls the ball just right so that it comes to rest in the inertial frame then when she completes one loop on the merry go round, the ball will still be there. It will have traced out a perfect circle from the point of view of the girl.

With any other launch speed or direction, the ball will follow a spiral path, not a circular one.

First, RaulTheUCSCSlug and jbrigs444, thanks for answering.

About the first question, again, I understood why the eastward defection takes place. What I didnt understand was why the eastward moving air, should suddenly move downward. Jbriggs44, you say that the eastward moving air experiences a force downward and upward, but I don't understand exactly what forces these are. Do all objects that move perpendicular to the axis of rotation experience a force upward or downward? If we rotate a ball by a thread it doesn't have any tendency to move downward or upward. I did some more research and came up with this article: http://stratus.ssec.wisc.edu/courses/gg101/coriolis/coriolis.html
In part C, it attributes the bending of air downward to the faster relative motion of the air (with respect to ground). Because of this faster motion the air tends to fly out of the atmosphere (higher velocity, more tendency to move in a straight line) but since it can't do that directly (courtesy of gravity) it goes towards the equator to increase the radius in any way it can. So is this the correct explanation/understanding of the concept?

Now about the girl/merry go round problem. As RaulTheUCSCSlug suggested, I drew the diagram with an eagle eye POV, frame by frame. And I took the motion of the ball to be a straight line. On observing with respect to the girl, I got a curve, but not a circle. So then again I took to googling, and searched for balls on rotating turntables and was surprised to find that a lot of research has already been done on this topic (I probably should have done this before asking this question itself, sorry) , even research that accounts for the role of friction. What seemed the most relevant was this link:
http://www.physics.umd.edu/deptinfo/facilities/lecdem/services/demos/demosd5/d5-11.htm [Broken]
Here they have rolled the ball and taken videos from a rotating camera as well as a stationary camera, and here again the path of the ball w.r.t the turntable was a curve but not a circle. I think the difference lay in the direction of the initial velocity of the ball. In the first video the girl rolled the ball in a direction that opposed the rotatory motion, much more so than in this link. Thus I am inclined to believe that as we change the direction of the velocity of the ball, more towards the opposing direction of rotatory motion, the curve slowly increases and finally becomes a circle. Does this sound about right?
Jbriggs444 you say that the ball must come to rest in the inertial frame and when she completes a loop the ball will still be there and will have completed a perfect circle. But this is not related to what is happening here I believe, because in your case, the ball should never move from the periphery of the disc, but in the video, it clearly moves toward the center and back to her hand. Also, if I am understanding what you said correctly, the other kids on the merry go round would collide with the ball in your case.

Also, another thing I'm confused about is the speed of the air packets. Does it ever change? In the video, it says that it doesn't, and of course, the coriolis force is fictitious, and therefore cannot change the speed of the air packets. The north/south moving packets' deflection towards east can easily be explained as due to the faster relative motion of the packets w.r.t the ground, and the east/west motion deflection is explained by inertia, and conservation of angular momentum never really comes into the picture. Of course, if it does come into the picture, speed will change as packets move closer and farther away from the axis. So does angular momentum conservation play any role here? Does speed change or not?

I see that I wrote upward and southward. I whould have written downward and southward since that is what a rightward deflection would produce.

Coriolis forces. Inertial forces that arise if you try to account for the apparent motion of objects when using a frame of reference that is rotating and pretending that it isn't.

I have not looked at the video. From what you say, the trajectory of the ball contains a closed loop. BUT NOT A CIRCLE.

You got a curve when you did the diagram, but if you do it in such a way where the ball reaches the other end the same time the girl reaches the other end, you actually get a circle. Try it out. Again, keep in mind your circular motion equation. (I did this for my lab) Sorry, but although I am convinced you did get a curve, it is possible to get it so that it makes a circle or at least an ellipse.

Neglecting friction (assuming the equivalent of a puck on an air hockey table), if the ball is thrown with a non-zero velocity in the inertial frame then it will move so as to eventually get further and further away from the axis of rotation. Its trajectory in the rotating frame can never be a circle or an ellipse in this case.

So I conducted an experiment. I cut out 8 circles from a transparent sheet. I took the ball to be moving with uniform velocity in a straight line across the diameter, and the girl to move half the circumference in the same time with uniform angular velocity. I took 8 different position of the ball and the girl ( when the ball is 0.25R across, 0.5 across and so on, and the girl has crosses pi*R/8, pi*R/4 and so on). Initially I lay the sheets on top of each other such that the ball is moving through the diameter in a dotted line, and the girl moves in the periphery. Then I rotated all the sheets and placed the girls dots one on top of the other (so that in this view, the girl is stationary and the other dots show the motion of the ball) and I got.... A perfect circle! I understand perfectly now. This problem is solved.