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From a quick google search:

Newton's first law states that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force.

This is simple enough and let's us know how to detect the presence of force.

Now consider a plane moving above the surface of earth(*), if we look at the path of the plane from space, it is true that the path of the plane is curved and hence there must be some force changing it's trajectory by first law. However, to the pilot in the plane it seems that it is moving straight line as everywhere is flat in the pilot's field of vision and therefore no force is acting on it by first law.

Therefore, what's straight in one perspective is not straight in another perspective. Hence, does newton's law fail when an object moves on a curved surface? If so, how do we fix it?


*: neglect rotation and revolution of earth

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The forces of gravity $F_g$ and lift $F_l$ act on the plane in opposite directions. $F_g - F_l = F_{cent}$ where $F_{cent}$ is the centripetal force keeping the plane moving in a circular path with respect to the earth. $F_{cent} = mv^2/r$ where $m$ is the mass of the plane, $v$ is its velocity, and $r$ is the distance from the center of mass of the plane to the center of mass of the earth. The plane moves at velocity v due to the thrust force from its engines countering the forces of "drag" from the air. [Note. The plane has insufficient velocity to allow it to remain at a stable $r$ without any lift; that is; it is not in "orbit" about the earth like the moon.]

In the non-inertial reference frame attached to the moving plane, the plane experiences a centrifugal force that balances the centripetal force and keeps the plane fixed in that frame.

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  • $\begingroup$ Hmm how would the pilot understand that he is an non inertial frame? When he looks through the window of aeroplane he'd see the earth as flat and him travelling in straight lines $\endgroup$ May 18, 2021 at 16:23
  • $\begingroup$ Secondly, how would you define 'straight' on a curved surface like earth? $\endgroup$ May 18, 2021 at 16:24
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    $\begingroup$ The pilot just experiences forces; the frame of reference (inertial or non-inertial) is just a framework for describing the forces. At constant altitude the plane is moving in a circular path but the radius of the circle is large so it appears the plane is moving straight to the pilot; like watching a ship sail over the horizon, it appears to move in a straight line but moves in a circle (as evidenced by the bottom of the ship disappearing before the top over the horizon). A true straight line path would be one changing in only one cartesian direction relative to the center of the earth. $\endgroup$
    – John Darby
    May 18, 2021 at 17:18
  • $\begingroup$ The pilot weighs very lightly less in the moving plane (due to the centrifugal force outward) but that is too small an effect to be noticed. Similarly, you weigh slightly less at the equator than at the poles due to the centrifugal force in a non-inertial reference frame attached to the rotating earth $\endgroup$
    – John Darby
    May 18, 2021 at 17:20

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