Faster Than Light? or Newton's Basic Laws are Wrong!

I was thinking about this problem in my first year of college. It is that we know.. No signal can travel faster than Light Speed and Newton’s Laws are valid in all reference frames i.e. When a force applied to a body, no matter how big is, from Newton’s first law the body will get an acceleration that is equal to a = F / m

Consider a body that is perfectly rigid i.e. neither incompressible nor stretchable. Assume that the body has a length of 5*10^8 meters which is more than one light year. It was also assumed that the experiment is done in vacuum so that there will not be any external effects. Now a force “F” is applied in “x” direction so that the body is displaced to a displacement of “dx” in x direction.

From Newton’s Laws of Motion,

Acceleration in x direction = Force / mass of the object (M)

                   a = F/M

The body is considered to be made of system of particles with infinitesimal mass “dm”. From the Centre of Mass Principle it is known that the path and the acceleration of the centre of mass is as same as the path of the point with mass “M” and acceleration “a”. Centre of Mass of an object that has uniformly distributed mass depends on the geometry of the object. We choose the object such that its centre of mass lies in that object.

From calculations we get,

Acceleration of Centre of Mass = F/M

So, it is sure that when it is applied an external force F to the object of mass M then the centre of mass of the object will get an acceleration of magnitude F/M.

From Electrostatics theory

When an external force is given to an object like rope, the motion of the rope is can be described as follows,

Consider that the rope has the molecules only arranged in linear order. Let’s say 5 molecules.

When the external force is applied, the first molecule will move in the direction of the force.

Let us take the case of pulling a rope and the Force Body Diagram of the first molecule is

Only if   “ Fext ” is greater than the net electrostatic attraction or repulsive force on the first molecule due to all other molecules namely -

“ Felectro-1 ”  , the first molecule will get an acceleration in the direction of “ Fext ” and will move distance "dx".

When the first molecule is moved through the distance “dx” in x direction, the second molecule will also move in the “x” direction to maintain the stable configuration. 

So the force on second molecule will increase to“ Fext ”  and it will be greater than the net attraction or repulsive force one second molecule due to all other molecules namely“ Felectro-2 ” . So, the second molecule will also move by distance “dx”.

Note: The body is considered to be perfectly rigid so that each molecule will be displaced by same distance “dx”. And so the external force will be transmitted to each particle.

In this way, throughout there will be electrostatic interaction between the molecules. And the interaction will make the last molecule that is at the other tip of the rope to move by the same displacement “dx”.

Thus we can explain the movement with Electrostatics Interaction.   

Here comes the question:

When the same situation is dealt with the object of such large length like 5*10^8 meters, it comes the problem.

The problem is,

From Special Theory of Relativity, “No kind of Electrical interaction can travel faster than Light Speed”.

But from Newton’s Laws of Motion, “The object that is the centre of mass of the object should instantaneously get acceleration

a = F/M and it is independent of time. The processes are instantaneous. ”. 

Which will happen in real life?

The question is whether the body will immediately move or not.

If it behaves by following the Special theory of relativity,

Then the other end of the object will move after some time interval of applied force. But it will be completely against the Newton’s laws of Motion. And it looks like a paradox.