Is there any frame of reference in which we can neglect the gravitational force?

Question

There is a laboratory in which there is a string tied to a bob on the earth.Consider the earth is perfect sphere ,rotates on its own axis in a circular path with uniform angular velocity.  Main Q of this concept is to find apparent weight of the object but I have some more Q related to this.
So , I tried to draw every possible frame of reference I can. Pls check if it is correct or not.

In earth frame of reference , I excluded the centripetal force and added a pseudo force. My confusion in this part is that if as an NIF , I look at the box from earth. Does it mean I am inside the earth on the ground or can I say I am a person running on the earth surface (I.e as if I am 1million feet in height )who has same acc as that of the box on the ground of earth or are both the situations same in meaning ?.
My Q 1:is that is it possible if we can take a non inertial frame due to which when I consider the FBD of the box , the mg or N force is not present.
It is more like I also wish  to know when do cos die the mg , N forces. I am not getting when to exclude them in NIF.

niels nielsen · Answer

The classic example of a frame of reference in which gravitational forces disappear is called a freely-falling reference frame.
This is what you get when in deep space, or in orbit around the earth, or when falling off a tall building or down a mine shaft on earth.
In a freely-falling reference frame, pendulums do not swing because all the (mg) terms vanish.

Semoi · Answer

In the so called center of mass frame the gravitational forces of the constituents sums to zero per definition. That's the reason why we describe two body problems as (1) motion of the center of mass, and (2) relative motion with respect to the cantor of mass.

kamran · Answer

it is called free float. And Einstein used it as an intuition leading to his theory of general relativity. In a small local spacetime no physical ebservation or experiment can tell the difference between weightlessness and free float.
the picture is from A Jurney into Gravity.. by John A Wheeler.
It is said that he considered this was one of his the happist thought.

Is there any frame of reference in which we can neglect the gravitational force?

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