Theory of Relative Circular Motion

An object that is in circular motion experiences more complex dynamics than one in a linear motion. The contrast between circular motion and linear motion is that particles experience different types of forces and their forces are radius-dependent with respect to a fixed axis within that body. In other words, an external force acting in a direction other that the original motion will result in a change in force sich that the force of any particle is different with respent to one another about a fixed axis within the same body.

An object experiences linear motion only if no external force acts upon it to shift from its original direction, The earlier chapter on Newton's Law of Inertia also states that a body at rest will stay at rest. What happens, then, when an object starts to rotate? In order to understand this situation beyond the centrifugal force that we have already discussed, we need to examine some general features of the object.

Let's begin by looking at a car that is traveling in a straight line. When the car reaches a bend (left turn), the driver makes a judgement and turns the wheel so as to follow the bend. If he decides that he wants to move straight, then he may continue to move straight and drive away from the bend of the road, which essentially means that he breaks away from the winding of the road. This however, by the virtue of his original state of motion, does not require extra force-assuming that there is no centripetal force acting on the car. In fact, by turning the car actually causes a change in velocity, which in turn causes an acceleration towards the center. This creates centripetal force and thus, centrifugal force. Centrifugal force is in this context the force to move in a straight line. In another words, the car has nowhere else to go but to follow the path of the road, which is a bend. The analogy in this context is that the car has an option of traveling forward or takes the bend. So if the car chooses to take the straight road, the force will only be a linear component . But if it chooses the bend, then there will be extra components of the forces governing the motion. These competitive forces will decide the ultimate resultant force for that motion.

If we expand our example a little, let's assume that it is a long road and the driver has no idea that it is bending. The road will appear to the driver to be straight, and the driver continues to drive in the same manner as before. Because the bend of the curve is so minimal, he hardly feels any of thecentripetal force that is acting on him. Together with the fact that the road appears to be straight, he will conclude that the road is straight. Little does he know that hew was traveling tangentially on a huge circle! Simply speaking, he is traveling on a straight road at the point of the curve. This is somewhat similar to our daily experience with gravity and the rotational motion that we are in. We don't really "feel" it even we know it is there. While this notion may seem ambiguous at this stage, it explains the first component of a particle in circular motion which is going to be helpful when we go further.

Next, we look at a particle (our car, again) in a restricted frame. Imagine the above scenario, but this time including a motorcycle in the inner lane traveling parallel to the car. As the car makes a left turn, the motorcycle, in order to avoid a collision (anti-collision), will have no choice but to make a left turn along the bend as well. This is what I mean by the rotational motion of a particle in a restricted frame. In contrast to our earlier discussion about the car's option of going straight or making the curve, a particle that is traveling in a solid state is not only constrained by the direction-as in the case of the car and the motorcycle-but also by the velocity that is traveling within that frame. Relatively speaking, in a rigid body-be it a rod or sphere- the velocity (angular or tangential) of a particle cannot be faster than another particle that is further than the former from the center of rotation. this should be true provided that, other thing being equal, no additional force is impressed on it.

Now let's consider what actually causes a car to turn. You can say it the wheel of the vehicle that allows the car to turn, which is partly true. But the real kinematics behind that turn is that the front wheels of the car have actually decided to take a different direction and a shorter path against the original path. By changing the angle of the wheel, the right side of the car will take a longer route and the left sided will have a shorter route to travel (see fig.2). However, the acceleration is the same for both sides of the front wheels. In such a case, the wheels will not be able to achieve a position of equilibrium at this point of time. As mentioned earlier in our anti-collision analogy, which will bring about a change of velocity in both set of wheels, which then causes the car to turn. If the turning os too acute, as in the case of a car traveling at high velocity, both wheels will not agree on the direction or else the car will be out of control, as different side of the car will try to take a different route with a different velocity. Fortunate.y, modern roads are made to bend in a gradual manner so that the frictional force between the tires and the ground is able to counter this centrifugal force and maintain stability while the car is turning. For discussion purposes, if we suppose that only one wheel is made to turn, leaving the rest of the wheels going forward, what will be the resultant motion of the car then?

The natural laws of rotation have the same direction if left to itself.


An object is said to be in unform circular motion when it travels in a circle at a constant speed. Although the speed is constant, the velocity continually changes as the direction changes. Hence, this motion is described as an accelerated motion and this acceleration tends to be directed toward the center. Why is this so when the centrifugal force seems to be directed outwards? This effect is caused by the change in the direction of the tangential velocity of the body in motion. In order for the direction of a body's motion to be changed, an external or internal force must be applied. Hence, the greater the change of the angle, the greater centrifugal force will be experienced. On the contrary, if the object's final tangential velocity about a center is less than the initial velocity, then the object will experience a deceleration.

Imagine being in a car and driving at a constant speed. Both your body and the car are in motion together. If you step on the brake, you will experience a force that pushes you forward. Actually, it wasn't really a force since there was no acceleration. It was the natural state of your motion that causes your to move forward while the car is slowing down. Relatively, speaking, the deceleration of the causes an equal acceleration on you and therefor, you felt a force. Likewise, while the car turns around a bend, your natural state of motion is straight while the car is angiualr and therefore, centrifugal force is "created".

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.

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.

The Natural Laws of Motion continue to have the same direction if left to itself.

Motion is a process of movement or behavior of an object that has traveled over a distance with time, as measured by an observer in a frame of reference. Only relative motion can be measured and is meangingful.

Motion can change the state of matter. When an object is at rest, we say it has mass. When it starts to move, we can measure its dynamics with terms like momentum, force, kinetic energy, etc. These are merely to describe the magnitude and effects of the motion. As the speed gets higher, the energy gets intensified. For instance, when water gets the right amount of heat, the molecules in the water will gain enough energy to escape in the form of steam.

Let us imagine an object of mass m at rest in space-motionless. We can say it has an inertia of mass m. First, we apply a little force on it, just enough for it to move at a constant velocity; then we can have what we call momentum=mv. If we continue to apply more force such that the object begins to accelerate, we will get to the next level as force, which is F=ma.

Hence, we can see how mass and motion can produce energy.

God had it all round.

There is a simple and yet powerful law observable in nature which governs the entire Universe. Remarkably, this law has been able to dictate all the natural laws of motion without flaw or variation. Otherwise, the slightest negligence would render the entire Universe into chaos!

Just think about the movement of a particle to the perpetual motion of the heavenly bodies. Birds can fly because their wings were to designed to lift them into the air, fishes can swim because they have fins to propel them through the water. There must be a set of rules that were written specifically for them. But who is capable of engineering such intricate designs and still able to execute them in such a complex Universe?

In order for something to have "life", that thing need to have "food" or a source of energy. Just as that thing need to have energy, it needs to moves. When it moves, that is motion. When it's in motion, it will have force or momentum, etc. Most importantly, there must be be enough space so that the thing can move for as long as necessary.

Consider, if the astronomical bodies were to travel in a straight path, how much of space would be required to accomodate them? Would life still exist if our mother Earth moves away from the Solar System?

Hence, the logical answer to these queries would be that everything has to be cyclical and conform to the Principles of Rotation.

The Principles of Rotation allows everything to travel in a manner prescribed within a certain space without disrupting other nearby bodies in their original state of motion. In other words, each particle can be in perpetual motion. Electrons can orbit an infinite of times, so does planets, stars, comets, etc.

Furthurmore, the Principle of Rotation allows the energy to be conserved or re-cycyled. Like a simple pendulum, energy is conserved in the simple harmonic motion.

The Laws of God are the same among themselves.

Life exists everywhere. We must not restrict the word "life" to just living things. The trees and plants have lives because they grow and bloom. The planets have lives because they orbit and create their own activities. Even a piece of metal has life because there are atomic particles inside that are constantly in motion.

By this definition, every form of matter has a life of its own. The difference is that these various forms exist differently and therefore behave differently. But they do behave, just like we have our own statutes and legislations to obey, they also have own laws to follow.

The most distinct thing about matter is motion. Everything moves. Tree grows, birds fly, celestial bodies circle in perpetual orbits, electrons spin and so forth. They obey the laws of nature, and any world without these laws to govern their movements would be in total chaos.