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Seems very simple, although not intuitive. But hey! If there is a single most often repeated statement in contemporary physics, it would be probably this one: “Intuition is not the final argument in science; science is about models, equations and predictions”. True. Inertial mass has been proven to be equal to gravitational mass, and therefore the force required to move inertial mass must exceed its force of gravity at the surface.

I hope, bobby, that you will find the answer to most of your question in this text.

Seems very simple, although not intuitive. But hey! If there is a single most often repeated statement in contemporary physics, it would be probably this one: “Intuition is not the final argument in science; science is about models, equations and predictions”. True. Inertial mass has been proven to be equal to gravitational mass, and therefore the force required to move inertial mass must exceed its force of gravity at the surface.

To sum up, addressing the original questions by bobby:

1. What is the (physics) difference between: 'inertia', 'force of inertia' and 'inertial force'?

There is none. All these terms express the same property of mass - its innate resistance to external force (acceleration). What makes inertia special is that it resists acceleration (force), and it takes another acceleration (force) to do that.

Also, following the line of reasoning above, which is confirmed by the famous Eötvös experiment, we can say that there is yet another synonym to inertia - gravitation. And following the Ockham's razor principle, it would only be logical to assume twin properties - inertia and gravity - to be simply one and the same thing.

2. Is it now just one of the fictitious forces or what?

If understood correctly, no. Inertia is the fundamental reason why it requires a real force to change the motion of a body. It's real, because mass, and nothing else, really resists a change to its motion. Also, inertia, being a synonym of gravity, is - as shown by Einstein - a real acceleration when measured at the surface of the body (source).

3. Is the concept of [force of] inertia still useful and used?

4. Can you list a few situations in which, if we didn't use this tool we might be in difficulty?

I take these two questions as provocative, or intending to explicitly demonstrate that dismissing (force of) inertia is not so wise an idea, to say the least ...

Whenever there is mass influencing the mechanics and equations of motion, there is always the concept of inertia involved. Because mass could (and should) be understood just as another synonym for inertia.

As to the situations where inertia - and therefore mass - cannot be neglected as a concept. There were some examples given in the comments (by Jim) to the question. There is a plenitude of examples: road traffic (car safety belts, bumpers, road railings, safety helmets), planes, lifts/elevators, or even building designs. So, in all situations, where mass (understood as acceleration out and therefore exerting real force on other objects it is in contact with) affects the equations and reality they describe, inertia is included by definition, and therefore dismissing it would get us into trouble. Even relativity that is believed to do just fine without (force of) inertia does provide equation for relativistic mass. Why? Because inertia is a fundamental factor when dealing with motion, and even particle physics must accept that.

Winding it all up - unless we manage to get rid of the concept of mass, we also cannot get rid of its synonym, inertia (or another synonym, gravity) - as simple as that.

The question of mass has arguably been one of the two most important issues in physics (the other being the electromagnetism). Physics has tried to uncover the true nature of mass for hundreds of years, to no avail so far. The description is rather circular:

Well, was it? Aside from the fact that acceleration is still measurable on the surface of the source of gravity, there is also another question left. Apparently, Einstein missed one thing: the geometrical space-time curvature concept, which replaced the force of gravity in GR, quite well explains the movement along geodesics, but does not explain the very impulse to motion in this field. The curvature by itself cannot make things move. If there is no force “underneath” the curvature, there is no reason the body should move at all. The usual counter-argument at this moment is that this is not a problem in GR, because under the concept of space-time and velocity 4-vector, each body is always in motion along the time axis. Assuming time is really orthogonal to space, Newton's first law of motion still says that one needs a force not only to set a body in motion, but also to change the direction of its motion. A body that moves along the $t$ axis requires a force to change the direction of its motion toward any of the space axes. And this change without a force is unexplained. This omission is really serious, since the ability to make things move is by far the most important aspect of mass and gravity.

Let's follow a simple line of logic then. What does it take to produce (a change in) motion? Acceleration obviously. So inertia resists acceleration. Now, what does it take to resist acceleration? Another acceleration. Would that suggest that inertia is acceleration itself? Well, each massive body generates gravitational field, and gravitation is acceleration. All seems to fit in.

How would that work in practice? If one is trying to move a material body, one must work against the body's own acceleration pointing outwards. It is a fact as long as Einstein's equivalence principle holds true. The more massive the body, the bigger the acceleration it produces, and the more external force it takes to counter this acceleration.

Seems very simple, although not intuitive. But hey! If there is a single most often repeated statement in contemporary physics, it would probably be this one: “Intuition is not the final argument in science; science is about models, equations and predictions”. True. Inertial mass has been proven to be equal to gravitational mass, and therefore the force required to move inertial mass must exceed its force of gravity at the surface.

The question of mass has arguably been one of the two most important issues in physics (the other being the electromagnetism). Physics has tried to uncover the true nature of mass for hundreds of years, to no avail so far. Not surprisingly, its description is somewhat circular:

Well, was it? Aside from the fact that acceleration is still measurable on the surface of the source of gravity, there is also another question left. Apparently, Einstein missed one thing: the geometrical space-time curvature concept, which replaced the force of gravity in GR, quite well explains the movement along geodesics, but does not explain the very impulse to motion in this field. The curvature by itself cannot make things move. If there is no force “underneath” the curvature, there is no reason the body should move at all. The usual counter-argument at this moment is that this is not a problem in GR, because under the concept of space-time and velocity 4-vector, each body is always in motion along the time axis. Assuming time is really orthogonal to space, Newton's first law of motion still says that one needs a force not only to set a body in motion, but also to change the direction of its motion. A body that moves along the $t$ axis requires a force to change the direction of its motion toward any of the space axes. And this change without a force is unexplained. This omission is a really serious one, since the ability to make things move is by far the most important aspect of mass and gravity.

Let's follow a simple line of logic then. What is a change in motion? Acceleration obviously. So inertia resists any acceleration. Now, what does it take to resist acceleration? Another acceleration. Would that suggest that inertia is acceleration itself? Well, each massive body generates gravitational field, and gravitation is acceleration. All seems to fit in.

How would that work in practice? If one is trying to move a material body, one must work against the body's own acceleration pointing outwards. It is a fact as long as Einstein's equivalence principle holds true. The more massive the body, the bigger the acceleration it produces, and the more external force it takes to work against this acceleration.

Seems very simple, although not intuitive. But hey! If there is a single most often repeated statement in contemporary physics, it would be probably this one: “Intuition is not the final argument in science; science is about models, equations and predictions”. True. Inertial mass has been proven to be equal to gravitational mass, and therefore the force required to move inertial mass must exceed its force of gravity at the surface.

Well, was it? Aside from the fact that acceleration is still measurable on the surface of the source of gravity, there is also another question left. Apparently, Einstein missed one thing: the geometrical space-time curvature concept, which replaced the force of gravity in GR, quite well explains the movement along geodesics, but does not explain the very impetus to motion in this field. The curvature by itself cannot make things move. If there is no force “underneath” the curvature, there is no reason the body should move at all. The usual counter-argument at this moment is that this is not a problem in GR, because under the concept of space-time and velocity 4-vector, each body is always in motion along the time axis. Assuming time is really orthogonal to space, Newton's first law of motion still says that one needs a force not only to set a body in motion, but also to change the direction of its motion. A body that moves along the $t$ axis requires a force to change the direction of its motion toward any of the space axes. And this change without a force is unexplained. This omission is really serious, since the ability to make things move is by far the most important aspect of mass and gravity.

Well, was it? Aside from the fact that acceleration is still measurable on the surface of the source of gravity, there is also another question left. Apparently, Einstein missed one thing: the geometrical space-time curvature concept, which replaced the force of gravity in GR, quite well explains the movement along geodesics, but does not explain the very impulse to motion in this field. The curvature by itself cannot make things move. If there is no force “underneath” the curvature, there is no reason the body should move at all. The usual counter-argument at this moment is that this is not a problem in GR, because under the concept of space-time and velocity 4-vector, each body is always in motion along the time axis. Assuming time is really orthogonal to space, Newton's first law of motion still says that one needs a force not only to set a body in motion, but also to change the direction of its motion. A body that moves along the $t$ axis requires a force to change the direction of its motion toward any of the space axes. And this change without a force is unexplained. This omission is really serious, since the ability to make things move is by far the most important aspect of mass and gravity.