Relativity - When something is said to be in motion, it must have its position changed in relation to other things around it as time progresses. This is the principle of the relativity of motion.
Force - A force is an quantification of some mechanical effect.
Particle - A particle is a body whose spatial dimensions can be neglected while describing its motion.
Aristotle’s Laws of Motion
1. An object with no unbalanced force will naturally come to rest.
2. An unbalanced constant force causes an object to move with a constant velocity.
3. Larger (more massive) objects apply larger forces compared to smaller (less massive) objects moving at the same velocity.
The above three statements can be reduced to the following statement: F = Rv where R is the resistance of the object against motion. Though Aristotle did not stated his laws like that but the above ones are pretty equivalent to his logic.
From the above laws we find an assumption made by Aristotle that a cause is necessary for state of motion and no cause is required for the state of rest.
Let us revise the principle of relativity as the new "postulate of relativity" which says that - All uniform motion is relative and all the laws of physics are the same for all such uniformly moving observers.
What is the rationale behind such a postulate? You will find the answer soon but let me start with the logical reasoning of Aristotle's laws together with the postulate of relativity.
IMPORTANT - Causality does NOT imply the postulate of relativity.
Consider some objects A and B.
A is at rest relative to you (the observer), so you both don't need any cause for this state.
B is moving at uniform velocity relative to you, hence it is moving relative to A as well.
According to Aristotle, B needs some cause of motion, I will call it "CM", to stay in motion.
But from the principle of relativity, both you and A are moving at some other uniform velocity relative to B.
So B can conclude that you need the CM to stay in motion, not B.
In the frame of A, A is at rest and B needs the CM to move.
In the frame of B, B is at rest and A needs the CM to move.
Now since the CM is some event itself hence to be consistent with the postulate of relativity, only one of the following statements has to be true:
Option 1. If the CM exists in the frame of A, then it must exist in the frame of B as well and vice versa.
option 2. If the CM does NOT exist in the frame of A, then it must NOT exist in the frame of B as well and vice versa.
Why? Because if the event CM exists only in one frame then the laws of physics would be different in the two frames.
But the laws of Aristole implies that:
Existence of CM for A (B moving relative A) implies non-existence of CM for B. (B is not moving relative to B)
Existence of CM for B (A moving relative B) implies non-existence of CM for A. (A is not moving relative to A)
So CM cannot have simultaneous existence in the frames of both A and B.
But what happens if CM does NOT exist in the frames of both A and B?
Since non-existence of CM in one frame does not imply the existence in another, we can say:
1. Both A and B do not need any CM to be in the state they are. The state of rest and the state of uniform motion are equally "cause-less".
2. But there may exists some causes of change in the states of uniform motions for both A and B.
3. If there exists such cause of change in the state for A w.r.t. B, then there has to be cause of change in the state for B w.r.t. A and vice versa.
The above philosophical analysis was mathematically formalized by Newton:
First - Law of Inertia - An object with no unbalanced forces will move at a constant velocity or be at rest at all times.
Second - Law of Forces - Unbalanced constant force causes an object to move with a constant acceleration. (F=ma)
Third - Law of Action and Reaction - An object is at rest or in uniform motion if there are no forces, but also when all the applied forces cancel each other. So for a composite system of at least two objects in mechanical equilibrium, the force on the first by the second has a reactive force on the second by the first.
Mass - Mass is an inherent property of matter that quantifies Inertia. Inertia is the property of matter by which matter opposes changes in its state of motion or rest. But what causes inertia is something I do not know yet.
Are the above the three laws of Newton dependent on each other? Yes, they reduce to "F = ma".
These laws were built upon the following postulates and they will explain you the rationale behind the postulate of relativity:
1. Space is infinite, continuous, homogeneous, isotropic, and 3-dimensional.
2. Time is infinite, continuous, homogeneous, 1-dimensional, and one-directional.
3. Space, time, and mass are independent concepts.
4. An inertial reference frame is a frame in which all previous postulates are satisfied.
5. All inertial reference frames can be transformed by Galilean transformation to all other inertial reference frames.
6. All laws describing a system are invariant within inertial reference frames.
An inertial frame is one such that no experiments, conducted completely within the frame, can ever tell us about the motion of the frame. It assumes a homogeneous and isotropic space, and homogeneous time. (Different locations and orientations don't affect the EoM and different instants of time also don't affect the EoM)
Whereas, in a non-inertial frame, we can easily detect motion without any reference to outside, just by conducting certain experiments confined within the reference frame. Space is inhomogeneous (different positions are mechanically different) and anisotropic (different orientations of the system are also mechanically different) and time is inhomogeneous (different instants are mechanically different).
To call space infinite means that the particular isolatable aspect of reality that we call space is of such a nature that it cannot define a limit to existent reality, and that, therefore, insofar as reality exists it is spatial. The infinity of time is significant in a perfectly analogous fashion. In summary, we do not know of any limits to space and time, so we better assume them as infinite. It works also since if something is moving at constant velocity, it must be moving eternally relative to an inertial observer, so the space and time must be infinite or eternal.
Since we haven’t found any gaps and jumps in space and time, we can also theoretically divide space and time into arbitrary units.
If space and time are not homogeneous, then one can find a special point in space and time that has different laws of physics, making the laws variable to different observers. Consider a particle moving along a straight line in one frame of reference. Under a non-linear transformation, this straight line would become curved, meaning that the particle is accelerated in that frame of reference, which cannot be, as the same laws are required to hold in all inertial systems. Hence, the inertial frame transformation has to be linear.
If space was not isotropic, then at least one direction would be preferred, and consequently, laws of motion in that direction would be different.
If time was isotropic, while causes always precede the effects (causality), some observer will see the opposite, making laws of physics different again for different inertial observers.
Think of a particle placed at a position defined by the position vector r0, and it has initial velocity v0, so we can always find the value of the next position, but how will we find the position after that? We need to know:
1. Initial position (space is infinite, continuous, homogeneous, isotropic, so objects can take any position).
2. Initial velocity (space is isotropic, so objects can move in any direction, so they have some velocities before the kinematics are calculated).
3. How the velocity changes after the initial moment of time. This is called acceleration and needs a cause called force. Hence, F = ma is a second-order differential equation.
An immovable object cannot be accelerated, and an unstoppable force means that the force agent always pushes or pulls normal things such that there is no effect on the velocity of the forcing agent. So the object imparting the unstoppable force also cannot accelerate. Both the unstoppable force carrier and the immovable object are the same thing, seen from different perspectives (both can’t accelerate means they always must be in uniform motion). Hence, when they try to collide, they must just pass through each other. As they can't collide.
A non-inertial frame can be realized as part of an inertial reference frame. Non-inertial reference frames are just a small part of either another non-inertial reference frame or an inertial reference frame. Formally, inertial frames of reference are philosophically more fundamental in classical physics. If you can come out of a non-inertial frame enough that you know what is the physical agent of the force that was unbalanced in the non-inertial frame, then you are in an inertial reference frame. If there is still some unbalanced force, keep doing this. But this won't end unless you take everything in your frame. Effectively, taking the entire universe, and since apart from the fundamental interactions, the universe is expanding, the universe is also a non-inertial frame. The movement of other galaxies tells us that we are moving too far away from each other at large scales of this universe...but why? And what is beyond this universe to explain this? We don't know.
For such a system, the F = ma will still be applicable. Here, m will be the total mass of all particles, and F will be the vector sum of all the forces acting on each particle, giving the acceleration of the entire system. This acceleration will not be the vector sum of the individual acceleration vectors but rather a mass-weighted average of those vectors.
The position vector described for a system of particles will be the mass-weighted average of the position of all particles. This vector point is where the mass is majorly concentrated, hence we call it the position of the center of mass (COM).
If there is no net force, F = 0, and the individual forces on each particle cancel each other (the mechanical equilibrium). then the acceleration of the center of mass will be zero. Hence, the velocity of the center of mass is constant. If there is a change in this COM velocity, then it must be due to some external force.
Consider a force that knocks off one or more particles out of this system. This surely changes the acceleration of the system, so it means that changing the total mass of the system is also an effect of an externally applied force. So Newton had to improve upon his F = ma equation a bit to make it general for a system of particles, not just one particle. While a force is meant to change the state of motion of a system, we find that in a composite system, it can change the number of constituents of the system, resulting in a change in the total mass. So force causes change in both velocity and mass of a system of N particles. Hence, we write F = change in m and v = change in mv (simplest mathematical relation for the AND logic).
Since this new quantity shows the state of the system undergoing translational motion, we call it translational or linear momentum. A vector denoted with p, and p = mv. And F = dp/dt is the Newton's law written much better. This tells us that for no external net force, a system of particles will conserve the total momentum of the system. One can also conclude that the velocity of center of mass of the system is conserved.
Each system is composed of particles, but these particles can be themselves a system, so it is better to stick with the concept of momenta even for indivisible particles. Hence, the positions and momenta of all particles define the mechanical configuration of the system.
Newton knew that objects that are left in the air tend to fall on the ground, the so-called gravitation. This observation makes the Earth's surface a non-inertial frame of reference whenever there is any vertical component in the motion involved. So why do objects stop at the ground, why don't they continue to fall? Because if you make a pit, the object will stop only at the bottom of the pit. With his laws, he deduced that some force must balance the force of gravity. This also explains why birds can fly and fishes can swim; the act of flying or swimming prevents things from falling as the medium somehow provides a force that balances gravity.
When an object is placed on the ground, it stays, so the force of gravity (weight) is balanced by a normal force provided by the ground itself. So if the same ground is seemingly attracting the object, how come it is pushing it upward? Also, is the normal force a cause of weight, or is the weight the cause of the normal force? What is applied to the object first? One reasoning is that the mechanisms responsible for the force of gravity must be different from the ones responsible for the opposing normal force. Also, it seems that gravity is more fundamental since sometimes some things get deformed as weight is applied to them from some other object from above. So the normal force is a responsive force caused by the weight. This makes us think about how gravitation works and what is the structure of matter. Such a problem requires a valid generalisation of Newton's laws.
Lastly, all the postulates regarding the universe requied for inertial reference frames and for the validity of Newton's laws are still questionable and require repeated observational confirmations. From all these started the classical physics, like a baby.