Does the centrifugal force exist in inertial frames too? Suppose I have a string hinged at one end and the other end attached to a body is to start a circular motion.
We all know that the tension force is just an aspect of the electromagnetic forces ( depends upon the separation between atoms in consideration). Here is the picture of the scenario. Note the gap between the string and the hook. Consider It as a highly magnified picture of the separation between them.


I took that string and when it was at rest with the body in a hanging position, the tension $T$ by the string (and also on the string) equals the $mg$ force on that body but the moment it completes first revolution and reached the initial point (i.e. at zero angle with the vertical) , it breaks !!!
What does that mean? It surely signifies that the body applied a greater force on the string and the molecular force in the string couldn't increase accordingly and hence it breaks.  And this is what mathematics of circular motion predicts :
$T = mg + \frac {mv^2}{r}$
Why did the body apply a greater force on that string?
I can only think that it applied a greater force because the molecules in the string were elongated somewhat but  the  cause of this elongation in the string is increase in force by the body on the string and this ultimately indicates that the molecules of the body came closer to that of the string.

This means that when we started circulating, the body was pushed out radially, and from an accelerating frame, the cause for this is said to be the centrifugal force but what if we observe it from an inertial frame.

What will be the cause of the body being pushed outward (or the separation between the molecules decreasing ; which is clearly visible in the above two pictures) from an inertial frame of reference? Which force is playing the role here? Does this experiment signify that centrifugal force also exists in an inertial frame or am I wrong somewhere?

Note: Notice the gap between the string and the hook in the first two images. I have drawn it accordingly.
Can't anyone give a reasonable point as none of the two answers give a proper physical reason?
Hope the question is clear now.
 A: The centrifugal force is a fictitious force, yes (a pseudo-force). It does not exist in an inertial frame. But the centrifugal effect does exist.
In fact, it is the other way around: The centrifugal effect exists, and then we in turn invent the idea of fictitious force to try to explain that effect to ourselves in accelerated frame.
The whole idea is that there are two things that can cause forces to appear:

*

*Other forces that must be balanced/counteracted

*Acceleration of a body with inertia that must be stopped/started

In your situation we have the latter case. The object is swinging as a pendulum and because of that it is constantly being turned towards the centre; a constant centripetal acceleration. The string causes this acceleration via its tension force, and via Newton's 3rd law the object applies that same force on the string. The string must now either

*

*elongate to allow for the object to move further down as the swinging tends towards (extending the distance between the particles of the string, meaning increasing the microscopic elastic forces),

*increase the force to match what is needed to cause the required centripetal acceleration at that swinging radius (requiring larger microscopic forces between all particles making up the string) or

*let go of the object entirely.

Since the string can't elongate (its too stiff) and since it isn't strong enough to apply the necessary force (the needed force exceeds the material's strength of the bonds between particles), the string breaks.
Nowhere in this description/analysis did we need the idea of a fictitious centrifugal force. It is accounted for in the inertia of the object; in the fact that the object is moving and a force is required to change this motion (to accelerated it; to turn the directino).
A: The centrifugal force is a pseudoforce.  It is not actually a real force.  Others have mentioned this of course, but no answer would be complete without it.
Now let's think about this problem in the inertial frame, while rotating.  You have an object in motion.  It wishes to stay in motion.  If no outside forces (electrostatic forces, in this case) were to oppose it, it would continue going in a straight line.  But you didn't want it to go in a straight line.  You wanted it to go in a circular path.  In order to do this, you use the string to apply a force to the mass.  You move the molecules in the string such that the electrostatic forces applied enough force on the mass to put it on a circular path.
How much force?  Well, this is a bit of a chicken and the egg problem.  The reality is that you constructed the experiment to cause circular motion to occur.  So its fair to think about this backwards, and start with the acceleration of the block.  If you accelerate an object towards a fixed point with a magnitude of $\frac{v^2}{r}$, where $v$ is the magnitude of the velocity of the mass, and $r$ is the length between your fixed point and the mass, the object will follow a circular path.  That can be proven using calculus and the geometry of the problem.  This means that you have constructed the experimental setup such that the electrostatic forces must apply $\frac{mv^2}{r}$, where $m$ is the mass of the block.
How you constructed this force is an artifact of the electrostatic effects pulling and pushing atoms.  Your application of force actually lengthen the string by just a tiny bit, and this lengthening increases the electrostatic forces.  We call it "stretching" in layman's terms.  In theory there is actually a complicated interplay between forces and stretching which can generate all sorts of effects, but for a simple string like we are talking about, what matters is that they occur over very small changes in length and generate rapid changes in forces, and that these effects stabilize at a given length and force.  We will handwave this away a fair bit here, but in real situations we do have to consider it.  In designing jet engine blades, the actual oscillating dynamics of this matters a great deal because we design those blades on the bleeding edge of what the materials allow.  For a closer-to-home example, this video shows a CD being spun too fast and torn apart.  If you watch just before it explodes, you can see the kind of odd dynamics that occur in this regime!
So now we know that the string is applying a force of $\frac{mv^2}{r}$.  The object tries to move in a straight line, and the electrostatic forces pull on it with a force that changes very rapidly with length, and stabilizes to where it is pulling, on average, with a force of $\frac{mv^2}{r}$.  If this force exceeds the strength of the electrostatic attractions, the string breaks.
Everything stated here is true in the inertial frame.  I did not need a rotating frame.  However, I did have to handwave a whole bunch of calculus.  There were all of the differential equations which dealt with the forces on the string and dealing with the fact that they were constantly changing in direction.  This is a pest.  It's mathematically accurate, but really annoying.
We can make the math simpler by viewing this in a rotating frame, whose rotation rate exactly matches the rotation rate of the object.  When we do this framing operation, there's one simple rule: the actual motion of the objects should not change.  This is intuitive.  We do not want the objects to actually take a different path, just because we thought about them different.  We may notate that path differently, but it should be the same physical path taken.
In this rotating reference frame, the forces are much simpler.  There are still electrostatic forces pulling on the string/hook/etc.  They are exactly the same electrostatic forces as they were in the inertial frame.  However, we are now thinking about them in a different way.  Now, rather than those electrostatic forces pulling in a constantly changing direction, we find that they are always pulling in the same direction -- radially.  This makes the math much easier.
However, an object in motion will continue in motion in a straight line.  But now we have the coordinate system rotating out of the way.  If we did nothing to change the equations of motion of the system, we would see, erroneously, that objects continue in motion along a circular path, which is clearly false unless they have a force (like electrostatics) pushing on them.  The answer is that we have to change the equations of motion in this rotating frame so that they describe the exact same motion as occurs in the inertial frame.  To do this, we add a centrifugal acceleration, $\frac{v^2}{r}$.  I'm pedantic about the acceleration bit because it isn't a force in the physical sense.  Its an acceleration term which needs to be considered to cause the motion in the rotating frame to coincide exactly with the motion we observed in the inertial frame.
Now we learn in physics that $\Sigma F=0$.  The  sum of the forces on an object equals zero.  It gets drilled into us, and it's wrong.†  $\Sigma F = ma$.  The sum of the forces equals the mass of the object times its acceleration.  If you got the second one drilled into you, consider yourself lucky.  You were taught well!
So, in the rotating reference frame, we have $F=m(\frac{v^2}{r}+a)$, that is to say that the sum of the forces will equal the total acceleration, which is the acceleration needed to account for the accelerations needed to correct the equations of motion to match what happens in the inertial world, plus some "visible" acceleration that we see by changes in position in the rotating frame.
So here's where the centrifugal force came in.  If you decide that you are going to think about this situation and forget that we are in a rotating frame, you will need a way to book-keep that acceleration.  To think about this rotating frame as-if it were inertial, you need $\Sigma F=ma$.  And to do that, we note that $F=m(\frac{v^2}{r}+a)$ can also be written $F - m\frac{v^2}{r}=ma$, which looks a while lot like an inertial system, but with this new "centrifugal force" term.  It exists only because we chose to forget that the rotating system's equations of motion had an acceleration in them.
This is where the centrifugal force came from.  It came from the decision to think about the problem as-if it were a non-rotating problem, and we had to book-keep the centripetal acceleration terms somehow.  The electrostatic forces are still the same in both frames, you still have the atoms in the string pulling on the hook, but we account for it differently in different frames.

*

*In an inertial frame, the electrostatic forces are causing the motion of the object to curve along a circular path.

*In a rotating frame, the electrostatic forces are opposing the centripetal acceleration term in the equations of motion, causing it to move in a path that has no radial component

*In a rotating frame that you are treating as-if it were inertial, the electrostatic forces are opposing this fictitious "centrifugal force," which is really just your way of book-keeping the accelerations that you chose to forget about.

†.This gets drilled into us as an unfortunate artifact of teaching.  Real dynamics problems, particularly interesting dynamics problems, almost always require lots of calculus and lots of effects.  Real statics problems (where $a=0$), even interesting statics problems, tend to be pretty easy to solve.  So a substantial portion of our interesting problems in class are statics, where $\Sigma F = 0$.  If the teachers don't make a big enough deal about this being only true for statics, its easy to internalize $\Sigma F =0$ and forget that it doesn't always apply.  And, of course, if teachers don't show us enough interesting problems, we start to question why we need Physics.  It's a bit of a Catch-22 for the poor teachers!
A: 
I took that string and when it is at rest with the body in a hanging position, the tension $T$ by the string (and also on the string equals) the $mg$ force on that body but the moment I oscillate the string, it breaks!
What does that mean? It surely signifies that the body applied a greater force on the string, the molecular force in the string couldn't increase accordingly, and hence it breaks.
Why did the body apply a greater force on that string? I can only think that it applied a greater force because the molecules in the string and the hook came closer and this is possible only if the hook ( or the body) is pushed outward i.e. towards the molecules of the string.

There's a simpler explanation, which is that when you yank the string, you're changing the acceleration on the body, which changes the force on the body due to $F = ma$.
Sufficiently abrupt yanking $\leftrightarrow$ large accelerations $\leftrightarrow$ large forces which exceed the tensile strength of the material that the string is made out of $\leftrightarrow$ the string breaks.
A: The situation is not much different if instead of starting a circular movement, we pull the string to an rectilinear movement.
In that case, there is a force at one of the ends of the string and the accelerated mass at the other. Tension in the string comes from its elongation: $\sigma = E\epsilon$. And it results from atoms or molecules having an average distance greater than the equilibrium.
The only difference of the circular movement is that we can not replace $\mathbf F = m\mathbf a$ by $|\mathbf F| = m\frac{d|\mathbf v|}{dt}$. The acceleration is normal to the velocity, and its direction points from the mass to where we hold the string, as happens in the rectilinear movement.
So, it is a centripetal acceleration and force.
A: The term "frame of reference" refers to a system of assigning numbers to events in space-time to describe "where" they are. Those numbers are called "coordinates". An inertial frame of reference is one in which an object with no forces acting on it will travel in a "straight line" in terms of the coordinates: that is, for each coordinate $c_i$, we have $c_i = mt+b$ for some $m,b$.

but what if we observe it from an inertial frame.

What will be the cause of the body being pushed outward


Suppose at time $t_0$ the string is pointing due North and the body is $1$ m outward from your hand. At time $t_1$ the string is $1$ degree NE and the body is $1.01$ m outward from your hand. You seem to be analyzing this as the mass moving $0.01$ m "outward". However, at time $t_0$, "outward" refers to an axis that is due North. At time $t_1$, "outward" refers to an axis that is $1$ degree off from due North. So with respect to the frame of reference of the Earth, your "outward" axis is constantly moving.
You are characterizing where the body is by how far "outward" it is, and you are thus using "outward" as a coordinate. But an object without any forces on it will not (unless it's moving directly towards or away from you) have its distance "outward" characterized by $mt+b$ for any $m,b$. So any analysis that described the body in terms of how far "outward" it is is not an inertial reference frame.
You are looking at what happens with respect to the end of the string. The body moves away from the end of the string, so you see the body being "pushed away". But the end of the string is accelerating, so looking at what happens from its perspective is not an inertial reference frame.
