The First Principle of Rotation

Rigid Body

The First Principle of Rotation occurs when a rigid body rotates, the tangential velocity of each point increases proportionally with the radius and unless the the radius of the point in the centre reaches zero, then any point other than zero will have an angular velocity itself.

Let us consider a rigid object, say a piece of metal rod, rotated through an angle about a pivot O. Imagine this rod is made of many particles. However, for simplicity sake, we will just take two particles on the rod and examine their kinematics in circular motion.

Let 2 particles, 1 and 2 sit on the rod, each having a certain radius r from the axis of rotation while 1 being in the middle and 2 being the furthest. If we measure the distances that they have covered after rotating through an angle after a certain time t, we will discover that both particles actually travelled a different distance relative to one another while the time t remained the same. In fact, all particles on this rod, which have a different radius from the axis of rotation would have covered different distances and therefore, have different velocities. As such, different velocities causes different accelerations. For this reason, it is not easy to fix an inertial frame of reference at any particular point other than the one is on the angular co-ordinate-the axis of rotation.

By the behaviour of such particles, let us describe this phenomena as position equilibrium between that of the particles. The particles traveling in a solid state are therefore not allowed to move any faster or slower than the speed that they are formulated to travel with respect to the other.

If we assume that particle 2 which is on the outermost of the rod to have the maximum tangential velocity, then we can deduce that particle 1 can only travel slower than 2. Theoritically, speaking, as we get nearer to the center, we can determine that the particle at the center of rotation will have an angular velocity close to 0 m/s. On the contrary, the said result is an illustration of the amplification of the magnitude of a particle in circular motion. If we assume the speed of light c is the ultimate speed that any particle in the Universe can travel, particle 2 will then achieve the speed of light so that particle 1 will obey the law and travel slower with respect to 2. There are two catches here. First, what if particle 1 is already moving at the speed of light, what will happen to 2 then? Analogously, if we impress a force such that particle 1 moves at the speed of light, what will be the speed of particle 2 then? Will it continue to move at the same speed or will it move faster or slower? Second, the context here is a restricted state of a rigid body. The dynamism will change once the system changes to a different state. This idea will discussed later.

Now that we have a little understanding of the first state of rotational motion, let's resolve one more issue before going any further. It is in the reader's best interest to have a comprehensive understanding about the relative motion of particles in an isolated system, regardless of their state of motion. If you put two particles in an isolated system and scrutinze them thoroughly, you will then be able to deduce and conclude the validity of the event through observations and experiments. One caveat: if you were able to treat the particles in an isolated system the same as the observer's reference frame, then the results would be dependent on the reference frame of the observer, which may invalidate them. The debate here is that if you "look" at the particles that are in motion from a different perspective, you will be able to understand the kinematics behind it. Your observations will be valid because you (your frame of reference) will not be affected (independent) by the laws that are gverning the observed refernce frame for rotary motions.

One classic example is how the Earth was once treated as the center of the solar system. This tradition of thougth goes back to the Greeks. during that time, Aristotle decreed that the logical position of the Earth was in the center of the Universe. It is quite easy to understand why such a theorem was accepted at period of history. Observation made were synonymous with the orbits of the Sun and Moon. But when early astronomers began to study the orbits of the rest of the planets, they behave in a peculiar way. Some of the planets were observed to slow down, stop, move backward and then forward. Scientists couldn't explain this strange phenomenon but were unwilling to accept that there were flaw in this theorem. In 1543, Nicolaus Copernicus published a book entitled, On the Revolution of the Heavenly Spheres, which shook the world. His new idea actually described the Sun as being the center of the solar system, similar to the model we have today. The moral of this story tells us that observations made in one reference frame have to be synonymous with another reference frame. It is good to study the behaviour of a particular object in motion by having a reference frame fixed within that object. However, experiments conducted in an enclosed space have to synonymous with an observer's reference frame outside that space so that the laws of physics will always be the same.

Here is another example. If two men are sitting in a car that is traveling at uniform speed, both will will see that everything is moving relative to them so their net speed (together with the inside the car) is zero (observer: they are moving at uniform speed). Let's assume that this is a vintage car with no speedometer and the windows are all enclosed. The two men have no way of telling whether they are moving or not. If they drop an apple to the floor of the car, that apple will still drop perpendicularly. It is only from the point of view of an observer outside the car, who indeed at uniform speed at uniform speed. Now let us imagine begin to accelerate, the relative speed is still the same between the two travelers and it is still the observer who can still tell the difference by the difference of the initial speed and the final speed.

The results taken from two different reference frames my yield somewhat different values but the derivations can still be the same. I will try to explain again. In an accelerated reference frame, the travelers may observe the same measurements as if the measurement were taken from an inertial reference frame. In other words, the travelers may not be able to tell whether their car is accelerating or not. It is the same as if the car was still moving uniformly (observer: they are accelerating). But there is a catch in this case. If one of them were to drop an apple, then they would notice a change from earlier case. This time, because the car is accelerating, the apple will drop slightly behind them. On the contrary, if the car is decelerating, a dropped apple will fall slightly in front of them. It was as though some mysterious force had been impressed on the apple that causes it to move in front or behind (observer: this effect is caused by the change in speed of the car while the speed of the apple remains the same at the point of release). These two men started arguing. One will claimed that we are moving faster the apple and so the apple fell behind us and vice versa. The other said that there was an external force on the apple that caused the apple to drop in front of behind (depending o the direction of the force) and when there is no force (other that the gravitational pull) then it will drop perpendicularly. Then one suggested maybe it is because the car is no longer moving in a straight line, it could be that the car is flying at angle such that the apple drops in a trajectory in the opposite direction of the motion. They are really confused this time, and so probably is the reader!

Let's make this situation even more interesting (or perhaps more confusing). Suppose this car is in a space now and it doesn't and it doesn't have seats, so that the two travelers will be both weightless and motionless. They both conclude that there is a fictitious force that keep them suspended in mid-air (observer: lack of gravitational force). Suddenly, the car moves forward and the travelers move backwards (equal and opposite) and they think it is an act of gravitational force (observer: the car moves forward). The the car stops abruptly and the travelers move forward this time and hit the other side of the car and they think it is the act of another external force (observer: car stopped but the two men continued in forward motion). Then the car begins to rotate with the pivot at the other end of the car so that the travelers still experience a force acting non them on the same side of the car (observer: act of centrifugal force). Can they tell the difference between the first situation and the latter? Imagine that the car stops rotating but this time, rotates with the travelers as the pivoting point. They might, by the motion of the side of the wall. But what if the car is in the shape of a sphere? Could they still tell the difference then? Clearly, they wouldn't be able to. They would not even notice if the car (sphere) turned upside down! In this case, how could they explain all the fictitious forces that are happening? It wouldn't be possible. Only the observer (and you) can see that the "fictitious forces" are the acts of the motion behind it.