Paradox of a Merry-go-round

Let's imagine we have a merry-go-round in an enclosed compartment and there are 2 men sitting on it, Mr Albert and Mr Stephen. Albert sits on the extreme side of the merry-go-round (near the circumference) while Stephen sits between Albert and the centre of the merry-go-round. We also assume that both of them sits at the same 12 o'clock position (from top view). We also have an external observer, Mr Isaac as our referee.

Now, we begin to rotate the carousel one complete revolution, clockwise. After the rotation, both of them is back to their original position-at the same time. If you measure the distance that Albert has covered and the distance that Stephen has moved, you would not been surprised that Albert has actually moved a longer distance than Stephen. Since Albert has moved farther than Stephen, then how did Albert arrive at the finish position at the same time with Stephen? If you ask any one of them (Albert and Stephen), they would have said they were all moving together. From their perspective, nothing has changed. In fact, they don't even know that they have been rotated(remember that they were in an enclosed space and there was no reference point)!

So what happened? Well, from Isaac's(an external observer) point of view, the answer to that question must be either;
1. Albert is moving faster than Stephen or
2. Stephen is moving slower than Albert.

So this leads us to the next question, "How did this happen when none of them made an extra effort to move faster or slower?". There was simply no additional force applied. I don't know what you think about this but to me, this is simply awesome. Here's why, if someone is moving faster or slower, the force or energy that they possess, is definitely gonna be different.

Newton stated, in formulations of his laws of motions, that the motions of bodies included in a given space are the same among themselves, whether that is at rest or moves uniformly forward in a straight line. Specifically, this means that if an experiment is performed in a train for example, all obsevations in the train will appear the same as if the train is not moving, provided the train is moving uniformly in a straight line.

In an analagous statement, Einstein defined the Principle of Relativity as

a mass m moving uniformly in a straight line line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K', provided that latter is executing a uniform translatory motion with respect to K. In restropect, an assumption that exists in a reference-body K, whose condition of motion is such that the Galilean law holds with respect to a particle is left to itself and sufficently far removed from all other particles moves uniformly in a straight line. With reference to K the laws of nature were to be as simple as possible. But in addition to K, all bodies of reference to K' should be given in this sense, and they should be equivalent to K for the formulation of natural laws, provided that they are in a state of uniform rectilinear and non-rotary motion with respect to K.

In order to note the significance of this issue, we examine another classical example of Albert and Stephen in a train. The train is moving at a constant velocity and both men are sitting together. Relative to each other, they are not moving. This is correct because even Isaac says so. When Albert starts to make an extra effort to stride down the train(towards the same direction that the train is traveling), then only Isaac and Stephen can see that Albert is moving faster than Stephen. If Albert decides to walk the opposite direction, then, he will be seen as moving slower by Stephen and Isaac.

So what's the difference between the men in the train and the men in the carousel? The differences are;
1.They were in rotary motion.
2.None of them could tell that they were moving faster or slower in carousel. Relative to each other, they were not moving. However, only an external observer could tell that Albert was faster with reference to the centre or Stephen was slower with reference to Albert.
3.There was no extra effort/force applied

Suppose we ask Isaac to sit in the middle of the carousel, he will also see that Albert and Stephen were moving at the same speed. When Issac sit in the middle, he will also notice that his orientation has changed from north(starting at 12 o'clock) to east to south to west and back to north again.

The analogy here is simple: unless the radius of object is zero, any point other than zero will have an angular motion.

Now we ask Albert (who is sitting near the circumference of the carousel) to face Issac (in the middle) and begin to rotate clockwise, completing one revolution. Albert's orientation would have been changed from south(starting at 12 o'clock) to west to north to east and back to south again. If we think about this carefully again, we will noticed that Albert has also made a "spin" of its own.

Hence, in the absence of an internal force, a point having a certain radius from the centre will have a tangential velocity about the centre of the system and at the same time an angular velocity about its own centre in the same direction of the rotation.


Let's argue the above from a different perspective. Let's rotate the carousel once more but this time, we shall ask Albert to face south throughout the rotation. Starting at the 12 o'clock position again, while the carousel is rotated clockwise, Albert has to rotate anti-clockwise in order to compensate the change in every angle. By the time the carousel is fully rotated, Albert has also completed one full anti-clockwise spin about his own centre. In such a case, his angular kinetic energy about himself and the tangential kinetic energy about the carousel would have been in opposite directions. Essentially, this would have opposed the natural laws of rotations.

Specifically, unless influenced by an external force, the natural law of rotation of a point will have the same direction as the natural law of rotation of the system.

It might be worthwhile to validate this point by doing a simple experiment. Suppose we tie a ball to a string and swing it horizontally around our hand. After many rounds, we release it and observe the motion of the ball. Common experience tells us that the ball will fly tangentially at the point of release, which is true. However, we are not able to observe whether the ball actually spins or not as the ball gets further away from us. In such a case, we should swing in a vertical manner and release it while it's going up so that gravity will act upon it and the ball will drop back for us to see. You should be surprised to see not only that the ball spins but the direction of the spin is exactly the same direction of the swing!

Finally, the important part of this principle is that each and every particle having a different radius from the centre of common system will experience different velocity and therefore, different kinetic energy.